Once Upon a Spacetime

by

Bradford Skow

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of PhilosophyDepartment of Philosophy

New York UniversitySeptember, 2005

Hartry Field

c© Bradford SkowAll Rights Reserved, 2005

Acknowledgements

Thanks to my advisors Gordon Belot, Cian Dorr, and Hartry Field, for their com-ments, suggestions, and criticisms, and for providing models of what good phi-losophy can and should be. Thanks to my parents—all of them—for supportingme through graduate school. And special thanks to Peter Graham for the frequentmeetings and for the discussion that led to the writing of chapter 4.

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Abstract

This dissertation concerns the nature of spacetime. It is divided into two parts. Thefirst part, which comprises chapters 1, 2, and 3, addresses ontological questions:does spacetime exist? And if so, are there any other spatiotemporal things? Inchapter 1 I argue that spacetime does exist, and in chapter 2 I respond to modal argu-ments against this view. In chapter 3 I examine and defend supersubstantivalism—the claim that all concrete physical objects (tables, chairs, electrons and quarks) areregions of spacetime.

Four-dimensional spacetime, we are often told, ‘unifies’ space and time; ifwe believe in spacetime, then we do not believe that space and time are separatelyexisting things. But that does not mean that there is no distinction between spaceand time: we still distinguish between the spatial aspects and the temporal aspectsof spacetime. The second part of this dissertation, comprising chapter 4, looks atthis distinction. How is it made? In virtue of what are the temporal aspects of space-time temporal, rather than spatial? The standard view is that the temporal aspectsof spacetime are temporal because they play a distinctive role in the geometry ofspacetime. I argue that this view is false, and that the temporal aspects are temporalbecause they play a distinctive role in the geometry of spacetime and in the laws ofnature.

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Table of Contents

Acknowledgements iv

Abstract v

Table of Contents vi

List of Figures ix

1 An Argument for Substantivalism 11 The Standard Argument . . . . . . . . . . . . . . . . . . . . . . . . 12 Characterizing the Spatiotemporal Structure of the World . . . . . . 43 The Embeddability Criterion . . . . . . . . . . . . . . . . . . . . . 74 Machian Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Account 1: One Mixed Fundamental Distance Relation . . . . . . . 116 Arguing for Purity . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.1 Clarifying the Premises . . . . . . . . . . . . . . . . . . . . 136.2 Defending the Premises . . . . . . . . . . . . . . . . . . . 15

7 Account 2: Pure Distance Relations . . . . . . . . . . . . . . . . . 188 Purity and Substantivalism . . . . . . . . . . . . . . . . . . . . . . 209 Modal Relationalism . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Modal Arguments against Substantivalism 271 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Defending The First Premise . . . . . . . . . . . . . . . . . . . . . 29

2.1 The Leibniz Shift Argument . . . . . . . . . . . . . . . . . 29

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2.2 Back to the Hole Argument . . . . . . . . . . . . . . . . . 342.3 Confusions About The First Premise . . . . . . . . . . . . . 362.4 Defending the First Premise . . . . . . . . . . . . . . . . . 382.5 Counterpart Theory . . . . . . . . . . . . . . . . . . . . . . 42

3 The Second Premise . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Determinism: Examples and Counterexamples . . . . . . . 433.2 Analyzing Determinism . . . . . . . . . . . . . . . . . . . 453.3 A Formal Definition . . . . . . . . . . . . . . . . . . . . . 48

4 Earman’s and Stein’s Argument . . . . . . . . . . . . . . . . . . . 515 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Supersubstantivalism 541 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Varieties of Supersubstantivalism . . . . . . . . . . . . . . . . . . . 553 Is Dualism the Default View? . . . . . . . . . . . . . . . . . . . . . 574 Cartesian Supersubstantivalism . . . . . . . . . . . . . . . . . . . . 595 Spatiotemporality . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Newtonian Supersubstantivalism . . . . . . . . . . . . . . . . . . . 647 Moderate Supersubstantivalism . . . . . . . . . . . . . . . . . . . . 678 The Modal Argument Against Supersubstantivalism . . . . . . . . . 69

4 What Makes Time Different From Space? 721 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 Spacetime Diagrams and Directions in Spacetime . . . . . . . . . . 733 Does Geometry Distinguish Temporal from Spatial Directions? . . . 784 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Laws of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 Testing Our Intuitions . . . . . . . . . . . . . . . . . . . . . . . . . 887 Symmetric Laws and Indeterminacy . . . . . . . . . . . . . . . . . 928 Laws That Govern in a Spacelike Direction . . . . . . . . . . . . . 949 Alternative Views . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.1 The Causal Theory of Time . . . . . . . . . . . . . . . . . 989.2 Three-Dimensionalism . . . . . . . . . . . . . . . . . . . . 99

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9.3 The Brute Fact View . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 103

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List of Figures

1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Newtonian Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 764.4 Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 824.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Chapter 1

An Argument for Substantivalism

1 The Standard Argument

Distinguish relationalism about motion from relationalism about ontology. Rela-tionalists about ontology deny that spacetime exists. In doing so they oppose sub-stantivalists, who affirm that spacetime exists.1 (When I use it without qualification,‘relationalism’ means the same as ‘relationalism about ontology.’) Relationalistsabout motion assert that all motion is the relative motion of bodies. In doing sothey oppose absolutists, who affirm that there are some states of motion that are notstates of motion relative to this or that material object.

I will work in the context of pre-relativistic physics. The standard argumentfor substantivalism in this context, going back to Newton, has two premises:

1Some philosophers claim that relationalists do not deny the existence of space-time, but merely assert that spacetime is a logical construction from the spatiotem-poral relations among material objects. (Forbes (1987) is an example.) I’ve alwaysthought that to say that x’s are logical constructions is just another way to say thatx’s don’t exist. Speaking in terms of ‘logical constructions’ obscures the debate.

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(1) Relationalism about motion is false.

(2) If relationalism about motion is false, then spacetime exists.

To argue for (1), substantivalists claim that no adequate physical theory is consistentwith relationalism about motion. Newtonian gravitational theory has a well-posedinitial value problem: there is a unique future evolution from any complete state ofthe world at a time.2 But no competing theory of gravitation that is consistent withrelationalism about motion has a well-posed initial value problem. And so, giventhe availability of Newton’s theory, no relationalist theory can be adequate.

The argument that no theory that is consistent with relationalism about mo-tion has a well-posed initial value problem has premises. First we suppose (i) thatour world, and all physically possible worlds, are worlds of massive point particles.Then if relationalism about motion is true, the history of the world is a history ofinter-particle distances. We also suppose (ii) that Newtonian gravitational theory isempirically adequate and a good guide to physical possibility. That is, we supposethat one of the solutions to the equations of Newtonian gravitational theory cap-tures the actual history of inter-particle distances; and that each solution captures aphysically possible history of inter-particle distances. Finally we suppose (iii) thatif relationalism about motion is true, then to give the complete state of the world ata time is to specify the distances between each pair of particles and the velocity ofeach particle relative to each other.

Given what we have supposed, there can be no alternative to Newtonian grav-itational theory that both is consistent with relationalism about motion and has awell-posed initial value problem. For any complete state of the world at a time—for any specification of the distances between and relative velocities of each pairof particles—there is more than one physically possible future evolution. Here isan example (from Newtonian gravitational theory supplemented with Hooke’s law).Consider a (physically) possible world that contains just two lead balls connectedby a spring. At some given time the distance between them is one meter and therate at which this distance is changing is zero. According to Newtonian mechanics,

2Setting aside problems with space invaders (Earman 1986) and collision singu-larities.

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there are at least two physically possible futures: either the two balls are rotatingaround their center of mass at such a rate that the ‘centrifugal force’ exactly bal-ances the force from the spring, and so they remain at relative rest for all time; orthey are not rotating, and so the spring collapses and then expands, and the initialtime was at the moment when the balls turn from moving apart to moving backtogether.3

(Absolutists diagnose this failure as follows: what relationalists about motioncall ‘the complete state of the world at a time’ is not complete. There are distinctinstantaneous states of the world that the relationalist cannot distinguish. In theexample, there are at least two ways to ‘complete’ the relationalist description ofthe ball-and-spring system, completions that differ over the value of each ball’sabsolute velocity.)

That is the argument for (1). The argument for (2) is shorter. If relational-ism about motion is false and there are some states of motion that are not states ofrelative motion, then spacetime exists. For the only way to define these absolutemotions is by reference to spacetime. (A particle is undergoing absolute accelera-tion at a time, for example, just in case its worldline is not tangent to a geodesic atthat time.)

Relationalists like Leibniz and Mach claimed that the first part of this argu-ment failed. But no relationalist produced a theory with a well-posed initial valueproblem. Until recently. Julian Barbour has developed such a theory.4

(So where does the argument that there can be no such theory go wrong?There are different ways to develop Barbour’s theory. Different developments re-ject different parts of the argument. On one way, relationalists reject (iii), and sayinstead that the complete state of the universe at a time is given by the distancesbetween and relative velocities of all particles and, in addition, a holistic propertyof the entire universe at that time (holistic because not reducible to the properties ofits parts), its angular momentum. (Relationalists who take this approach say some-

3I first heard this example from David Albert.4This theory is discussed in detail in (Barbour 1999), (Belot 2000), (Butterfield

2002), (Pooley and Brown 2002), among other places. Belot notes that theories likethis were discovered as early as 1924.

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thing about why we do not need to believe in spacetime in order to make sense ofangular momentum.) On another way, relationalists reject (ii), that Newton’s theoryis a good guide to physical possibility. They deny that worlds in which the entireuniverse is rotating, including one where two lead balls connected by a spring arerotating around their center of mass, are physically possible.)

With theories like Barbour’s available, substantivalists are in a weaker posi-tion. No longer able to argue for (1) by pointing out that relationalists about motionhave no physical theory, they must now argue that absolutist theories like Newton’sare better than relationalist theories—and not better because more adequate to theempirical phenomena, for both theories do just as well on that score, but better forsome more subtle reason. But it is not clear that the absolutist theory is better:Barbour (1999) and Pooley and Brown (2002) argue that the relationalist theory isbetter than Newton’s because it predicts a phenomenon that in Newton’s must betaken as a brute fact: namely, the fact that the universe is not rotating.

To argue for substantivalism I take a different approach. In the debate overthe standard argument, it is widely assumed that we need spacetime to make senseof absolute acceleration, but that we do not need spacetime to make sense of thedistances between and relative velocities of material objects.5 I think this is wrong.Even if relationalism about motion is true, and we can do physics while recognizingonly distances between and relative velocities of material objects, we still needspacetime (I will argue) to make sense of the distances between material objects.Relationalism about motion demands substantivalism just as much as its denial.Any adequate characterization of the spatiotemporal structure of the world—evenone consistent with relationalism about motion—must make reference to spacetime.Before outlining my argument in any detail, I will explain this terminology.

2 Characterizing the Spatiotemporal Structure of the World

Relationalism about motion is an answer to the question:

(3) What is the spatiotemporal structure of the world?

5Sklar (1974) disputes the first half of this claim.

4

Both substantivalists and relationalists must answer this question, though they willnot answer it the same way. It is easiest to see what a complete, detailed answer tothis question will look like if we look at some substantivalist answers first.

Substantivalists believe in spacetime. Just as there are many possible ge-ometries that space may have (it may be Euclidean, or hyperbolic, for example),there are many possible geometries that spacetime may have: it may have a neo-Newtonian structure, for example, or a full Newtonian structure. When substanti-valists disagree about the answer to (3), then, they are disagreeing about which ofthese geometries the spacetime in our world has.

A substantivalist might answer (3) by saying that spacetime has a Newtoniangeometry. How might a relationalist answer (3)? She denies that spacetime exists,so she cannot give the same answer. But she does not think that (3) is an emptyquestion, and she might even think that, except for their ontological disagreement,the substantivalist’s answer is right.

To see the general form a relationalist answer may take, look more closelyat a substantivalist answer. Suppose some substantivalist says that spacetime has aNewtonian geometry. We may ask: in virtue of what does it have that geometry?And the answer is: it has that geometry because the points of spacetime instantiatecertain spatiotemporal relations in a certain pattern. Which spatiotemporal rela-tions? There are many possible answers to this question. Here is one: spacetimehas a Newtonian geometry because the points of spacetime instantiate the spatial

distance between x and y is r and the temporal interval between x and y is r in acertain pattern. (That is, there are geometrical laws that these relations satisfy.)

A relationalist who says that the world has a Newtonian spatiotemporal struc-ture might say, then, that it has this structure (at least in part) because the materialobjects, rather than the points of spacetime, instantiate certain relations in a certainpattern. (Maybe the relationalist uses the same relations the substantivalist does;maybe not.)

So both substantivalists and relationalists produce lists of spatiotemporal rela-tions and laws governing those relations in order to characterize the spatiotemporalstructure of the world. Now, for any given spatiotemporal structure the world mighthave there are many different sets of relations that characterize it. Euclidean space,

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for example, may be characterized using the relation the distance from x to y is

r; or using the relations y is between x and z and x and y are as far apart as z

and w; and there are other choices as well. But I am interested in which relationsare, according to substantivalists and to relationalists, the fundamental relations thatcharacterize it. Fundamental relations, I assume, have the following feature: factsabout the instantiation of non-fundamental relations obtain in virtue of the factsabout the instantiation of the fundamental relations.6 Facts about the instantiationof the fundamental relations, on the other hand, are brute, ‘bottom-level’ facts. (Thenon-fundamental relations, then, supervene on the fundamental relations. Since itis also true that the fundamental relations supervene on themselves, every relationsupervenes on the fundamental relations.)

Now I can outline my argument in more detail. I will argue that even if thecorrect answer to (3) entails relationalism about motion, relationalist accounts ofwhich spatiotemporal relations are fundamental are objectionable. So we shouldprefer substantivalism to relationalism.7

In my argument I impose a constraint on which spatiotemporal relations maybe fundamental. To state this constraint I need to introduce some terminology. Saythat relations which relate abstract objects to concrete objects (x is n years old isan example) are mixed relations; other relations—relations that relate only abstractobjects, or only concrete objects—are unmixed, or pure. The constraint is this:

Purity: Necessarily, the fundamental spatiotemporal relations are pure.8

6Here and throughout I limit this claim to qualitative, non-modal relations.7Again, I am working in the context of pre-relativistic physics. But the argu-

ments generalize to relativistic spacetime structures.8If one believes that all possible relations exist necessarily and that necessarily,

if a relation is fundamental then it is necessarily fundamental, then the modal op-erator at the beginning of Purity is redundant. In what follows I will make theseassumptions.

Purity presupposes that some fundamental relations are spatiotemporal relations.Leibniz denied this; resemblance nominalists (who think that the fundamental rela-tions are relations of resemblance) presumably deny this; some advocates of stringtheory or of some other theory of quantum gravity may deny this. My argumentsneed not presuppose that such views are false. Even if no fundamental relation is

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Here is an outline of the rest of this chapter. In section 3 I state a necessary condi-tion on relationalist characterizations of the spatiotemporal structure of the world,and in section 4 I explain in detail what the spatiotemporal structure that corre-sponds to relationalism about motion looks like and how it differs from Newtonianand neo-Newtonian spacetime. In section 5 I look at the most obvious relationalistcharacterization of this spatiotemporal structure and reject it because it is inconsis-tent with Purity. I argue for Purity in section 6, and in the remainder of the chapterI look at relationalist accounts of the fundamental spatiotemporal relations that arecompatible with Purity. I argue that they are unacceptable.9

3 The Embeddability Criterion

Suppose a relationalist answers (3) by saying that the world has a Newtonian spa-tiotemporal structure—not a likely answer, but this is just an example—and that hetells us a story about the fundamental facts in virtue of which it has this structure(a story that, as I said, will include a list of the fundamental spatiotemporal rela-tions that material objects instantiate and laws that those relations obey). I claimthat this relationalist story is correct only if it guarantees that relationalist worlds inwhich it is true are uniquely embeddable in Newtonian spacetime, up to the symme-tries of that spacetime. (And similarly for other possible spatiotemporal structuresthe world might have.) More precisely: let R4 with the spatial distance betweenany two points (x1, x2, x3, x4) and (y1, y2, y3, y4) given by the standard Euclidean for-mula ds(x, y) =

√(x2 − y2)2 + (x3 − y3)2 + (x4 − y4)2 and temporal distance given

by dt(x, y) = |x1 − y1| be our canonical model of Newtonian spacetime geometry.Then the relationalist laws are correct only if there is a one-to-one function f from

spatiotemporal, still (I take it) there are some spatiotemporal relations that are mostfundamental: no spatiotemporal relation is more fundamental than they are. Puritymay then be cast as a principle about the most fundamental spatiotemporal rela-tions. In the body of this paper I will ignore this complication, since nothing turnson it.

9The argument I give here is similar to the one Field gives in ‘Can We Dispensewith Spacetime?’ (1989). While I think that much of what I say is in the spirit ofthe arguments Field gives, there are important differences between our arguments.I will mention these differences in the relevant places.

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the pointsized things in the relationalist world into R4 such that

• f preserves spatial and temporal distance: for any pointsized instantaneous10

things x and y in the relationalist world, ds(x, y) = ds( f (x), f (y)) and dt(x, y) =

dt( f (x), f (y)), and

• every other such function differs from f by a symmetry of Newtonian space-time.

(I use ‘ds’ to name both the spatial distance function defined on the material ob-jects in the relationalist world and the spatial distance function defined on pointsof R4.) The symmetries of Newtonian spacetime are the one-to-one functions fromspacetime onto itself that preserve spatial and temporal distance. There are similardefinitions of symmetry and embeddability for other spacetimes. (Not all of thesedefinitions need be given in terms of a spatial and temporal distance function.)

Of course, the relationalist’s story may not explicitly mention a spatial ortemporal distance function. But these functions must be definable from the story hedoes tell, for certainly there are facts about the spatial and temporal separation ofany two pointsized instantaneous objects in a Newtonian relationalist world (givena choice of units of measurement), whether or not those facts are fundamental.11

The embeddability criterion is accepted by many who discuss the debatebetween relationalists and substantivalists.12 I take it that it needs little defense.Clearly, if a relationalist says that a world has a Newtonian spatiotemporal struc-ture, but there is no way to embed that world in Newtonian spacetime, then he is

10Relationalism is easiest to defend if is conjoined with the doctrine that everymaterial object is composed of pointsized instantaneous parts. (So this doctrineentails the doctrine of temporal parts.) If this doctrine is true, then to specify thespatiotemporal relations among the material objects in the world it is enough tospecify the spatiotemporal relations among all the pointsized instantaneous materialobjects.

11In fact it is possible to formulate the arguments in this paper in terms of thedefinability of a Euclidean spatial distance function in relationalist worlds, insteadof in terms of the embeddability of (parts of) relationalist worlds into Euclideanspace. I think framing the arguments in terms of embeddability makes the issuesclearer.

12(Belot 2000), (Earman 1989), and (Friedman 1983) are three examples.

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lying. And if the embedding is not unique up to a symmetry of the spacetime, thenthe relationalist’s story underdetermines the spatial relations among the materialobjects. (For example, if the relationalist says that the world has a Euclidean spatialstructure, but there are three instantaneous pointsized material objects, all simulta-neous, and there are two embeddings of them into Euclidean space, one which mapsthem to the vertices of an equilateral triangle, and one which maps them to the ver-tices of a triangle that is not equilateral, then the relationalist has not adequatelycharacterized the spatial structure of the world.)

Embeddability is a necessary condition that a relationalist characterization ofthe spatiotemporal structure of the world must meet. But it is not sufficient. Thereare two commonly cited obstacles to its sufficiency.13 First, any relationalist worldthat is embeddable in a four-dimensional Newtonian spacetime is also embeddablein a five-dimensional (or higher) Newtonian spacetime with an extra spatial dimen-sion. The embeddability criterion alone does not fix the dimensionality of space.Second, any relationalist world with (as a substantivalist would say) large enoughempty regions of spacetime that is embeddable in a four-dimensional Newtonianspacetime is also embeddable in a four-dimensional spacetime in which space isEuclidean in all the occupied regions, but is non-Euclidean in some of the emptyregions. So the embeddability criterion alone does not always fix the geometry ofunoccupied regions of spacetime.

I mention these problems with treating the embeddability criterion as suffi-cient only to set them aside. I will only appeal to its status as a necessary condition.

4 Machian Spacetime

To say that all motion is relative motion of bodies is to say something about thespatiotemporal structure of the world. I can now be more specific about just whatthis structure looks like. (To do so I speak as a substantivalist.) Earman (1989) listssix possible (non-relativistic) spacetimes and orders them by how much structurethey have. All the spacetimes Earman lists consist of a stack of three-dimensionalEuclidean instantaneous spaces. The order in which they are stacked gives their

13(Earman 1989).

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ordering in time. Machian spacetime is the weakest of these: all there is to thatspacetime is a stack of three-dimensional Euclidean instantaneous spaces. We cansay how far apart in space points on the same instantaneous space are, but nothow far apart in space or in time points on different instantaneous spaces are. Soin Machian spacetime the only meaningful questions are questions about how farapart particles are at any instant, and qualitative (not quantitative) questions abouttheir relative motion: whether two particles are getting closer together, for exam-ple. Machian spacetime, then, is a spacetime in which all motion is the relativemotion of bodies.14 Barbour designed his relationalist replacement for Newtonianmechanics to live in a Machian world. (By contrast, Newtonian spacetime has morestructure: it comes with a way to determine how far apart in both space and timepoints on different instantaneous spaces are. In this spacetime, we can tell whichtrajectories through spacetime are trajectories of particles at absolute rest, and wecan ask whether a single particle is in absolute motion, and if so what its numericalspeed is.)

Can relationalists say what it is for the world to have a Machian spatiotempo-ral structure?

To guarantee the embeddability of the world’s particles into Machian space-time relationalists must guarantee that each ‘time-slice’ of the world (each equiv-alence class of the world’s pointsized instantaneous material objects under the si-multaneity relation) is embeddable in three-dimensional Euclidean space. It mightseem at first like this is easy to do: all a relationalist needs to do specify the spatialdistances between the particles in that time-slice, and ensure that those distancesobey Euclidean laws. Then the time-slice will be uniquely embeddable into Eu-clidean space.

But what are the fundamental spatial relations that give the distances betweenthe particle-slices? There are several choices here, and the problems relationalistsface do not come out until we are more clear about which choice we have in mind.

14It is not the only such spacetime; Leibnizian spacetime, which differs fromMachian spacetime only by having a temporal metric, is also a spacetime in whichall motion is the relative motion of bodies.

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5 Account 1: One Mixed Fundamental Distance Relation

One account of the fundamental spatial relations a relationalist might use to accountfor the distances between particles is this: say that there is one fundamental spatialrelation, the distance from x to y is r. Then state laws for this relation that guaranteethe embeddability (up to uniqueness) into Euclidean space of the particle-slices in-stantiating it. I suspect that when relationalists think that a Machian spatiotemporalstructure is perfectly acceptable they do so because they think that this is all thatneed be done.

The problem with this account is that it is incompatible with Purity:

Purity: Necessarily, the fundamental spatiotemporal relations are pure.

It is now time to argue for this principle.

6 Arguing for Purity

There is one group of philosophers who will find Purity appealing: nominalists,those who deny that there are any abstract objects.15 For if Purity is false, because(say) the distance from x to y is r is the only fundamental spatial relation, and ifin addition there are no numbers, then no spatial relations are instantiated, and sothe world is not spatial at all. Since nominalists do not think their view entails thatspatiality is an illusion, they will embrace Purity.

But I think that even anti-nominalists should accept Purity. I myself find thefollowing argument convincing: even granting that there are numbers, the followingcounterfactual is true:

(4) If there were no numbers, the world would still be spatial, and in fact thespatial structure of the world would be just as it actually is.

But if Purity is false because the distance from x to y is r is the only fundamentalspatial relation, then (as before) if there were no numbers no spatial relations would

15Some nominalists deny that there are any properties or relations. They willdeny that Purity is literally true. But nominalists should have some fictionalistreading of sentences containing property talk, and they will accept Purity whenread that way.

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be instantiated. And any world in which no spatial relations are instantiated is aworld that is not spatial at all. And that means that (4) is false.16

Many anti-nominalists will be unconvinced by this argument, though. Theywill say that numbers exist necessarily, and so that (4) is a counterfactual with anecessarily false antecedent. So either (4) makes no sense at all, or it is vacuouslytrue; in either case, there is no problem for an anti-nominalist who accepts Purity.

So I will take a different approach. My argument for Purity is this:

(P1) If Purity is false, then the way the world’s particles are spatially arranged atany one time is not intrinsic to the particles.

(P2) The way the world’s particles are spatially arranged at any one time is intrin-sic to the particles.17

(C) So Purity is true.18

This argument is addressed to relationalists. I mean to convince relationalists toaccept its conclusion. (I think substantivalists should also accept its conclusion, butfor different reasons.19) I do not claim that its premises are true (its second premisein particular), just that relationalists should believe them. Before I explain why, Iwill clarify what the premises mean.

16If Purity is false because some, but not all, fundamental spatiotemporal rela-tions are mixed, then it does not follow that if there were no numbers the worldwould not be spatial. But in this case (4) is still false: if there were no numbers thenthe spatial structure of the world would not be just as it actually is.

17From now on I omit the reference to time.18Field also tries to motivate something like Purity by noting that a theory in-

compatible with Purity does not give intrinsic explanations of phenomena (1989,pages 192-193). I take it that an intrinsic explanation is one that appeals only to theintrinsic properties of and relations among the things invoked in the explanation; somy focus on the intrinsicness of the way the world’s particles are arranged is not, atbottom, different from Field’s focus on intrinsic explanations. But Field just takes itfor granted that (P1) is true. I have found many people willing to deny it; I providean argument for it.

19Substantivalists should believe that properties characterizing the geometricalstructure of spacetime (like the property of having a Newtonian geometrical struc-ture) are intrinsic. Arguments similar to the ones I give below show that if Purityis false, then these properties are not intrinsic.

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6.1 Clarifying the Premises

I use ‘the way the world’s particles are spatially arranged’ as a name of a relation.I take it that talk of the way some particles are spatially arranged is familiar: oneway for three particles to be spatially arranged, for example, is for any two of themto be as far apart as any other two. Then they stand at the vertices of an equilateraltriangle. In this case, then, ‘the way the particles are spatially arranged’ names thethree-place relation x and y are as far apart as y and z, and x and y are as far apart

as x and z. (P2) is the claim that relations like this are intrinsic. But what does itmean to say that a relation is intrinsic?

We are familiar with the distinction between intrinsic and extrinsic properties.Intuitively, an intrinsic property is a property that characterizes something as it isin itself. What intrinsic properties something has in no way depends on what otherthings exist (things other than it or its parts) or how it is related to them. Withextrinsic properties (properties that are not intrinsic), by contrast, other things can‘get in on the act’ when it comes to determining whether something instantiatesthem.

Although it is unfamiliar, the notion of an intrinsic relation is a straightfor-ward generalization of the notion of an intrinsic property. Just as an intrinsic prop-erty characterizes something as it is in itself, an intrinsic relation (with more thanone argument place) characterizes some things as they are in themselves. For ex-ample, x is as massive as y is intrinsic (or, at least, seems intrinsic at first): if twothings are equally massive, that is a matter of how those two things are in them-selves. Whether they instantiate x is as massive as y does not depend on what elsethere is or what those other things are like.20

20If x is as massive as y is intrinsic, then it is plausible that it is also internal.Internal relations supervene on the intrinsic properties of their relata. More for-mally, a relation R is internal just in case: if x and y instantiate R and x′ and y′

are duplicates of x and y (respectively) then x′ and y′ also instantiate R. (x′ and y′

need not exist in the same possible world as x and y. The generalization to relationswith more than two argument places is straightforward.) All internal relations areintrinsic, but not all intrinsic relations are internal. (Fundamental relations (withmore than one argument place) are intrinsic but not internal. Some define ‘externalrelation’ as ‘relation that is intrinsic but not internal.’ (Lewis (1983) for example.)

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Here is another way to think about intrinsic relations. Intrinsic relations usu-ally correspond to intrinsic properties of fusions. Consider the property x has two

parts that are equally massive. This property is intrinsic. Any intrinsic duplicateof something with two equally massive parts would itself have two equally massiveparts. And something instantiates this property just in case it has two parts thatinstantiate x and y are equally massive. And the latter (as I said) is an intrinsicrelation. In general, if an n-place relation R is intrinsic then the property of havingn parts that instantiate R is also intrinsic.21

Using the distinction between fundamental and non-fundamental relations Ican give a more precise characterization of ‘intrinsic relation.’ All relations super-vene on the fundamental relations. But this is global supervenience: if some thingsinstantiate a relation R, they do so in virtue of the global pattern of instantiation ofthe fundamental relations. An intrinsic relation, by contrast, supervenes ‘locally’on the fundamental relations. That is, a relation is intrinsic only if it supervenes onthe fundamental properties of, and fundamental relations among, its relata and itsrelata’s parts.

Local supervenience is necessary for a property to be intrinsic. But if there areany things that exist necessarily it is not sufficient. Suppose God exists necessarily;then the property of coexisting with God supervenes on any set of properties. Butit is not intrinsic.

The solution is to demand more: a relation Rx1...xn is intrinsic just in case itcan be analyzed in terms of the fundamental relations x1...xn and their parts instan-tiate. An analysis is a kind of ‘definition’ of the non-fundamental relation: to givean analysis is to display an open sentence that expresses that relation (and so hasas many free variables as that relation has argument places) such that every predi-cate in that open sentence expresses a fundamental relation. We can tell whether arelation is intrinsic by looking at its analysis: Rx1...xn is intrinsic just in case every

The distinction between internal and non-internal relations will not play a role inmy arguments.

21I believe the converse is true for fundamental relations, but can fail for non-fundamental relations. The (non-fundamental) relation ∃z (x and y compose z) isan example: it is not an intrinsic relation, but the property of having two parts thatcompose something is an intrinsic property.

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quantifier in its analysis is restricted to x1...xn and their parts.22

6.2 Defending the Premises

I have now explained what the premises mean, and given a more precise character-ization of ‘intrinsic relation.’ I turn now to arguing that relationalists should acceptthe two premises.

Here is my argument for (P1). Suppose Purity is false. Suppose in additionthat there is just one fundamental spatial relation, the distance from x to y is r. Thenthe way the world’s particles are spatially arranged may be analyzed in terms of thisrelation.

If this is true, then intuitively, the particles get to be spatially arranged in acertain way in virtue of being related in the right way to numbers. So their beingarranged in that way is not a matter of the way they are, in themselves; so the waythey are arranged is not intrinsic.

To go through this argument in more detail, fix on a particular way the par-ticles might be arranged, and look at its analysis. Suppose there are only threeparticles and that at a certain time they stand at the vertices of an equilateral trian-gle. Then the way they are arranged may be analyzed as follows:

∃r(the distance from x to y is r & the distance from y to z is r & the distancefrom z to x is r)

This analysis contains a quantifier that ranges over something other than x, y, z, andtheir parts; namely, a quantifier that ranges over numbers. So the relation analyzedis not intrinsic.

Someone might object that numbers (and abstract objects generally) shouldreceive a special dispensation in the definition of ‘intrinsic.’ That definition bans

22The definition of ‘intrinsic’ I am working with is related to the one given byDavid Lewis in On the Plurality of Worlds (1986b). It follows from Lewis’s defi-nition that the fundamental properties and relations are intrinsic. So this definitionis appealing only if we accept this consequence. But I don’t think this consequenceshould be controversial. It is part of our conception of fundamental properties thatthey are intrinsic: it is usually said that the fundamental properties make for simi-larity among their instances, and that they carve nature at its joints (Lewis 1983).

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quantifiers not restricted to the parts of the things instantiating the relation fromanalyses of intrinsic relations. But perhaps it should allow quantifiers that rangeover only abstract objects into such analyses; so long as no quantifier ranges overconcrete things other than the parts of the things instantiating the relation, the rela-tion is intrinsic. (The above analysis can easily be rewritten so that the first quanti-fier is restricted to numbers.)

We should reject this revised definition of ‘intrinsic.’ To illustrate why it iswrong, notice that it leads to incorrect results about the intrinsic properties of num-bers themselves. If we suppose that numbers exist, then it is plausible to supposethat x is the successor of y is a fundamental relation that natural numbers instantiate.(Other relations among the natural numbers—like addition and multiplication—canbe defined in terms of successor.) Now the property of being the smallest naturalnumber, like the property of being the shortest person in the room, is not intrinsic.Whether a number has this property depends on what other numbers there are, andwhether it is the successor of any of them. But this property’s analysis—‘¬∃y(yis a number and x is the successor of y)’—contains just one quantifier restricted tonumbers, and so the revised definition of ‘intrinsic’ says that it is intrinsic.

In this argument for (P1) I’ve assumed that if Purity is false then the distance

from x to y is r is the fundamental spatial relation. It should be clear from my discus-sion that nothing turns on using this particular mixed relation. The argument worksequally well no matter which mixed relation we choose. This is important because,if one were going to chose a mixed spatial relation to regard as a fundamental rela-tion, one would not be likely to choose the distance from x to y is r. It has seemedto many that this relation isn’t really fundamental, but is analyzed in terms of someother fundamental mixed spatial relations. For example, some have thought that thefundamental spatial relation concerns distance ratios, rather than distances. That is,there is just one five-place fundamental spatial relation, x and y are r times as far

apart as z and w. Distances between points are then analyzed in terms of the ratiosof their distances to the distance between two reference objects (say, the ends ofthe standard meter). This theory is appealing because is does away with seemingly-mysterious facts about which things are distance one unit apart. (We are very goodat determining when two things are one meter apart, and when they are one foot

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apart; but these facts look like facts about the ratio between the distance betweenthe two things in question and the distance between the endpoints of the standardmeter, or the standard foot. Asked to determine whether two things are one unitapart, in some absolute sense that is independent of any standard of measurement,and we do not know what to do.)23 But for all this view’s virtues, my argumentfor (P1) works just as well if it (or some other theory of mixed fundamental spatialrelations) is true.

So much for (P1). What about (P2)? Not everyone will accept (P2). Somesubstantivalists, in particular, will reject it. For some substantivalists say that theway the world’s particles are spatially arranged is derived from facts about wherein space each particle is located.24 The way the particles are arranged, then, is notjust a matter of how those particles are, in and of themselves, but is also a matter ofthe way they are related to space. So their spatial arrangement is not intrinsic.

I claim that relationalists should accept (P2). For (P2) is, I think, one of theclaims that motivates relationalism in the first place. Distinguish two motivationsfor relationalism. (There may be others.) One way to motivate relationalism is tocomplain that points of space and instants of time are unobservable entities, andassert that for that reason we should not believe in them. But another (and better)way to motivate relationalism is to complain that space and time are irrelevant andunnecessary. As I said in the previous paragraph, one role that space plays forsubstantivalists is to ‘ground’ spatial relations among material objects. I may bea mile above Los Angeles; according to substantivalists, what it is for me to be amile above Los Angeles is for me to be located at a certain region of space, and LosAngeles to be located at a certain region of space, and for those regions to be a mileapart. That is, substantivalists say that material objects inherit their spatial relationsfrom the spatial relations among the regions of space they occupy. But this detourthrough spatial relations among regions of space looks unnecessary. Why can’t the

23After reading Kripke (1980) you might doubt that ‘x and y are one meter apart’expresses the relation x and y are as far apart as a and b where a and b are the endpoints of the standard meter. I think the view that best fits with Kripke’s picture isaccount 2 (section 7 below; see also footnote 25).

24Not all substantivalists say this: supersubstantivalists, for example, do not.Supersubstantivalism is the subject of chapter 3.

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‘one mile apart’ relation hold directly between me and Los Angeles? Why does itneed to be derived from spatial relations among regions of space? And if it doesnot need to be so derived, then space is irrelevant. The way the world’s particles arespatially arranged is not a matter of the way they are embedded in space; instead,it is a matter of how those particles are, in and of themselves. It is intrinsic to theparticles. Here, for example, is Barbour expressing this motivation:

What is the reality of the universe? It is that in any instant the objects init have some relative arrangement. If just three objects exist, they forma triangle. In one instant the universe forms one triangle, in a differentinstant another. What is to be gained by supposing that either triangleis placed in invisible space? (1999, pp.68-69).

(Not for nothing does Barbour call his theory ‘intrinsic particle dynamics.’) I thinkthis, and not appeals to problems with unobservable entities, is the best motivationfor relationalism. It is a motivation that I myself feel, even though I am no rela-tionalist. But—what is important for my purposes—this line of thought motivatesrelationalism by appealing to (P2). So relationalists should accept (P2).

There is, of course, another well-known motivation for relationalism: theLeibniz shift argument. But I do not think that this motivation is very good. (Idiscuss this argument, and it successor the hole argument, in more detail in the nextchapter.)

I have argued that everyone should accept (P1) and relationalists should ac-cept (P2). So relationalists should accept (C): they should accept Purity.

Let me be clear about what this means. If numbers do exist, then materialobjects surely do bear mixed relations like the distance from x to y is r to numbers.I only claim that it cannot be relations like these that give the world its spatialstructure. When it comes to the fundamental facts about spatial structure, numbersare strictly irrelevant.

7 Account 2: Pure Distance Relations

Relationalists cannot characterize the spatial structure of the world by saying thatthe three-place spatial relation the distance from x to y is r is fundamental. Here

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is an alternative theory: say that there are infinitely many two-place fundamentalspatial relations. These relations have names like ‘the distance from x to y is one,’‘the distance from x to y is two,’ and so on. But we should not be misled by thenames: although one relation’s name contains the letters ‘o-n-e,’ we are not to thinkof these as a separate syntactic unit that serves to name a number. We are not tothink of these pure relations as derived from the three-place relation the distance

from x to y is r by filling in its third argument place with a number.25

If these relations are fundamental then the way the world’s particles are spa-tially arranged is intrinsic. So this theory is compatible with Purity and respectsthe motivation for relationalism. And so long as the inter-particle distances do notviolate Euclidean laws, the world’s time-slices will be uniquely embeddable intoEuclidean space.

The problem with this theory is that it just cannot be true. For according tothis theory there are necessary truths that we must think are brute and inexplicable,but which we ought to be able to explain. For example, it is necessary that if twothings instantiate the distance from x to y is one then they do not also instantiate the

distance from x to y is two. And not only is this necessary; each distance relationexcludes the other distance relations in this way. Or again, it is necessary that ifa and b instantiate the distance from x to y is two and b and c also instantiate the

distance from x to y is two then a and c do not instantiate the distance from x to y

25This seems to be the view Melia prefers at the end of his (Melia 1998). It alsofits best with Kripke’s discussion of the standard meter (1980). Kripke suggests thatwe refer to the standard meter stick only to fix the reference of ‘one meter apart.’Two things can be one meter apart without bearing any relation to the endpoints ofthe standard meter; they can be one meter apart even in possible worlds in whichthe standard meter stick does not exist. Account 2 provides the right sorts of dis-tance relations to be the semantic values of expressions like ‘one meter apart,’ asKripke understands them. So does a version of account 1: if the fundamental spa-tial relation is the three-place relation the distance from x to y is r then ‘one meterapart’ expresses the derived two-place relation the distance from x to y is n for somenumber n. But as I mentioned above, while on this second interpretation we can (ifasked) tell which things bear the distance from x to y is r to the same number that theendpoints of the standard meter actually do, if asked which things bear this relationto the number one, we have no idea how to find out the answer.

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is five. And not only is this necessary; other instances of the triangle inequality aretrue as well.

The problem is not that these necessary truths are not logical necessities. I donot claim that all necessities involving fundamental relations are logical (as someversions of the combinatorial theory of possibility do). The problem is that thesenecessities exhibit a striking pattern, and we ought to be able to explain why theyexhibit this pattern. But we cannot.26

This argument generalizes to other attempts to turn theories of mixed funda-mental spatial relations into theories of pure fundamental spatial relations by sub-stituting infinite families of pure n − 1 place relations for a single n place mixedrelation. For example, a theory that says that relations like x and y are two times as

far apart as z and w are fundamental fails for the same reasons.

8 Purity and Substantivalism

So far I have been using Purity to beat up on relationalists. But (as I remarked infootnote 19), substantivalists should also accept Purity. It may fairly be demandedwhether and how substantivalists can produce a theory of the fundamental spatialrelations that is compatible with Purity.

Substantivalists have no problem doing this. There is a synthetic axiomatiza-tion of Euclidean geometry using just two primitive predicates of points of space,‘x, y are congruent to z, w’ and ‘x is between y and z.’ (Actually there is onesuch axiomatization for two-dimensional Euclidean geometry, and one for three-dimensional Euclidean geometry, and so on; let’s stick to the three-dimensional

26Field argues that we would not want to accept a physical theory that used two-place predicates like ‘the distance from x to y is one’ as semantic primitives, be-cause it would be unlearnable (since it contains infinitely many primitive predicates)and unusable (since it contains infinitely many axioms that cannot be captured ina finite number of axiom schemas). But this is not enough to show that we cannotaccept that two-place relations like the distance from x to y is one are fundamen-tal: for the proponent of this metaphysical view may deny that the physical theoryscientists learn and calculate with needs to have semantic primitives that expressfundamental relations.

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case.) These two predicates clearly express pure spatial relations.27 Field (1980)has produced a similar synthetic axiomatization of the geometry of neo-Newtonianspacetime. A substantivalist in search of a Purity-friendly account of which re-lations are fundamental will be attracted to synthetic axiomatizations of geometrylike this. He need only accept that these synthetic axioms are true of spacetime andmake the further claim that the predicates which appear as semantic primitives inthe theory express fundamental relations.28

But this account of the fundamental spatiotemporal relations is not availableto relationalists. The first problem is that it is not obvious what laws the relational-ist will propose for betweenness and congruence. He cannot use the same laws thatsubstantivalists use. Those laws entail, among other things, that there are infinitelymany points of space; so if a relationalist accepted them and rewrote them to men-tion particles rather than points of space, he would have to accept that if the worldhas a Euclidean spatial structure then there are infinitely many particles. And norelationalist wants to accept that.

Relationalists could weaken the laws, only requiring that no set of particlesinstantiates (at a time) betweenness and congruence in a pattern that is contrary tothe substantivalist laws. This law amounts to a guarantee that there is at least oneembedding of any time-slice of a relationalist world into Euclidean space. (Whetherthis law can be stated directly, without mentioning embeddings, is a good question,but I set it aside.)

The problem now is that there are (in general) too many embeddings of theparticles into abstract Euclidean space. Here is an example.

Suppose that relationalism is true and that there are only three particles A, B

27There is a theorem that establishes that every model of these synthetic axiomsis isomorphic to R3 with a Euclidean geometry. This theorem is only true for thesecond-order axiomatization; Tarski sketches a proof of a similar result for the first-order theory in (Tarski 1959).

28And this is an additional claim. If he does not make it, then it is consistent withall that has been said that the unmixed relation expressed by ‘x, y are congruent withz, w’ is analyzed in terms of mixed fundamental relations. (Perhaps it is analyzedas there is an r such that the distance from x to y is r and the distance from z to w isr.) But then Purity is false.

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and C, and the facts about the instantiation of betweenness and congruence at a time(other than the trivial ones29) are as follows: betweenness is nowhere instantiated;congruence is nowhere instantiated. Any embedding of this relationalist world intoR3 maps these particles to the vertices of a triangle that is not an isosceles or anequilateral triangle; and for any such triangle there is an embedding that maps theparticles to the vertices of a triangle like that one. So, for example, there is anembedding of the particles that maps them to the vertices of triangle displayed infigure 1.1 on the left; and also one that maps them to the vertices of the triangledisplayed below on the right. Clearly, no isometry of Euclidean space maps one of

Figure 1.1:

these triangles onto the other. So on this version of relationalism, the embeddingof this world into Euclidean space is not unique up to isometry. And this is not aproblem with this particular choice of pure fundamental spatial relations; the sameproblem will arise if relationalists use some other choice of primitives for syntheticgeometry as a guide to which spatial relations are fundamental.30

9 Modal Relationalism

The betweenness and congruence theory has a lot going for it. It escapes the prob-lems that face both of the previous accounts I discussed. Why not use modality to

29The trivial facts are those like A is between A and A and A, B are congruent toA, B, the universal generalizations of which are theorems.

30Other choices of primitives are: congruence alone; the three-place predicate‘x, y, and z form a right triangle at y’; and the three-place predicate ‘the distancefrom x to y is less than or equal to the distance from y to z.’ For each choicethere are distinct spatial arrangements of three particles that are isomorphic withrespect to the primitive predicates. (Royden 1959) surveys choices of primitives forsynthetic geometry.

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patch up the relationalist version of the betweenness and congruence theory and se-cure the unique embeddability of time-slices into Euclidean space? After all, somerelationalists characterize the view in modal terms to begin with: Leibniz, the arch-relationalist, wrote: ‘space denotes, in terms of possibility, an order of things thatexist at the same time’ (Leibniz and Clarke 2000, page 14).

The idea is to define distance relations in terms of betweenness, congruence,and a modal operator, and to ensure that these distance relations obey Euclideanlaws. Then the world’s time-slices will be uniquely embeddable into Euclideanspace.

One way to do this is to look at how a substantivalist who accepts the be-tweenness and congruence theory can define four-place pure distance-ratio relationslike x and y are twice as far apart as z and w, and try to convert those into modaldefinitions. Substantivalists can define this relation in terms of betweenness andcongruence as:

(S 2) ∃v(v is between x and y and x and v are as far apart as z and w and v and y

are as far apart as z and w)

A relationalist might simply prefix this definition with a modal operator:

(R2) ♦∃v(v is between x and y and x and v are as far apart as z and w and v and y

are as far apart as z and w)

There is an analogous definition (R3) for x and y are three times as far apart as z

and w, and so on.There are two related problems with this approach.First, there are problems with the meaning of the modal operator. It cannot

express logical or metaphysical or physical possibility. It is logically and metaphys-ically and physically possible for the distance ratio between two pairs of particlesto be anything you please. If the modal operator expresses logical or metaphysicalor physical possibility, then, two given pairs of particles will satisfy both (R2) and(R3). But there must be a unique distance ratio between the pairs of particles.

It cannot express metaphysical possibility restricted to worlds in which thenon-modal spatial relations among the four points are as they actually are. Fixing

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the betweenness and congruence relations fixes all the non-modal spatial relations.Suppose there are only four particles and that no particle is between any of theothers and no two are as far apart as any other two. It is (metaphysically) consistentwith this description that the first two particles are twice as far apart as the secondtwo. It is also (metaphysically) consistent with this description that the first twoparticles are three times as far apart as the second two. So the four particles againinstantiate both (R2) and (R3), which they should not.31

Relationalists can say that the modal operator expresses ‘geometrical possi-bility.’32 But what is geometrical possibility? It must be some restriction of meta-physical possibility. I’ve looked at three restrictions; none of them work. I can thinkof no other candidate descriptions of this restriction. We may wonder whether weunderstand what this operator means at all.

There are also problems with necessary connections. I rejected account 2because it asked us to accept necessary truths involving distance relations as bruteand inexplicable. This account faces the same problem. Why is it that four particlescan only instantiate one of (R2), (R3), (R4), and so on? We cannot derive thesenecessary truths from the way the modal operator used is analyzed, since it has noanalysis.

The following is the best way for relationalists to respond to these problems:

I admit that I cannot ‘define’ the geometrical possibility operator byfinding some sentence and then saying that the geometrical possibilityoperator acts like a quantifier over just those metaphysically possibleworlds in which that sentence is true. But I can tell you somethingabout which worlds (in addition to the actual world itself) are geomet-rically possible relative to the actual world.

Among the metaphysically possible worlds are worlds in whichMachian spacetime exists, all and only the actual particles exist, andthe non-modal spatiotemporal relations among the particles are as theyactually are. For reasons given above, these worlds do not all agree on

31Field (1989) makes these points, and extends the argument to all spatial rela-tions, modal or not.

32(Belot 2000).

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the distance rations between pairs of particles at each time. But the setof these worlds divides up into equivalence classes, where members ofany one equivalence class do agree on the distances between the par-ticles at each time. One of these equivalence classes is special. Themembers of that class and no other are geometrically possible relativeto the actual world. But I can’t tell you which class is special, or whyit is special and the others aren’t.

(Which worlds are geometrically possible varies from world to world,and since there are worlds that are non-modal duplicates—have thesame history of betweenness and congruence relations—but differ indistance ratios, there are worlds which are non-modal duplicates butdiffer over which worlds are geometrically possible.)

This helps with the problems I mentioned: we know how the geometrical possibilityoperator works, and the necessary connections between distance ratios, while stillbrute, are not so mysterious.

Still, we should reject this proposal. For one thing, it asks us to believe inbrute modal differences: worlds which are isomorphic with respect to betweenness,congruence, and temporal ordering, and so are the same, as far as their non-modalfacts are concerned, but which differ over distance ratios—which are modal facts—between pairs of particles.

But the more important problem with this proposal is that it looks like cheat-ing. Relationalists need to guarantee the unique embeddability of the world’s par-ticles into Machian spacetime; the solution here is to use a modal operator to do itby brute force.

If this is how relationalists about motion hold on to relationalism about on-tology, then what is attractive about relationalism about motion in the first place?There are supposed to be epistemic and (related) metaphysical advantages: inter-particle distances and relative velocities are easier to observe, and a theory thatrelies only on them does not need to postulate spacetime. But if we can use thesestrange modal operators, then relationalism about motion no longer has these ad-vantages over its denial. As for metaphysical advantages: let the world have aNewtonian spatiotemporal structure, so that talk of absolute states of motion makes

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sense. We can still be relationalists; there is no need to appeal to spacetime tomake sense of absolute motion. Just use a new modal operator to guarantee theunique embedding of the world’s particles into Newtonian spacetime, and use thisembedding to define states of absolute acceleration. As for epistemic advantages: ifinter-particle distances and states of absolute acceleration are equally funny modalfacts, why should one be easier to observe?

Like accounts 1 and 2, modal relationalism is a bad move. Even if relation-alism about motion is true, relationalists cannot adequately characterize the spa-tiotemporal structure of the world. We should be substantivalists.

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Chapter 2

Modal Arguments againstSubstantivalism

1 Introduction

Two famous arguments against substantivalism—the Leibniz shift argument andthe hole argument—turn on substantivalism’s (alleged) modal commitments. Bothcontain premises to the effect that substantivalists must believe that it is possiblethat things be just as they are, qualitatively, while differing in some non-qualitativerespect.1 Leibniz argued that these the possibilities conflict with some a priori meta-physical and theological principles. Defenders of the hole argument argue that thepossibilities in question lead to a failure of determinism. Responding to the holeargument is harder, so I will focus on it. I will say something about how we shouldthink about Leibniz’s argument in the course of responding to the hole argument,though. The hole argument looks like this:

1Qualitative properties are those that ‘make no reference’ to particular individ-uals. So the property of being red is qualitative, while the property of being in thesame room as Ralph Nader is non-qualitative. If we have a language the predi-cates of which express qualitative properties, and which contains no proper names,then we cannot distinguish between qualitatively indiscernible worlds using thislanguage: every sentence true in one of the worlds is also true in the other. I saymore about what the qualitatively indiscernible worlds at work in the Leibniz shiftargument and the hole argument look like below.

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(1) If substantivalism is true, then (assuming that general relativity is the truetheory of the world2) it is physically possible that everything be just as itactually is, except that some spacetime points in the (absolute) future ‘playdifferent roles.’

(2) If there is more than one physically possible future consistent with the waythings are now, then determinism is false.

(3) Therefore, if substantivalism is true (and general relativity is the true theoryof the world), determinism is false.

I’m going to take it for granted (as most do) that accepting the conclusion—thatgeneral relativity is not a deterministic theory—is an embarrassment for substanti-valists. So there are two ways for substantivalists to respond to this argument. Theycan deny the existence of the qualitatively indiscernible possibilities mentioned inthe first premise; or they can deny the second premise, and so deny that such possi-bilitties lead to a failure of determinism.

I advocate the second kind of response. Making this response work involvesgetting clear on just what determinism is. I defend an analysis of determinism thatblocks the hole argument and is independently plausible. Although I defend thisanalysis of determinism in the context of responding to the hole argument, its valueis not limited to the use to which I put it. Understanding determinism is some-thing worth doing for its own sake. And, more importantly, determinism shows upelsewhere in the debate between relationalists and substantivalists. For example,Earman (1989), following Stein (1977), argues relationalism about motion entailsrelationalism about ontology. In the previous chapter I argued that relationalismabout motion requires substantivalism. Unless relationalism is inconsistent, wecannot both be right. Earman’s argument contains a premise about determinism;it presupposes the same analysis of determinism as the one at work in the hole ar-gument. So like the hole argument, Earman’s argument fails for relying on (whatI argue is) an incorrect analysis of determinism. I briefly discuss this argument in

2There is some debate over whether the hole argument applies to theories otherthan general relativity. Earman and Norton (1987) argue that it does, while Earman(1989) argues that it does not. This controversy won’t matter for my discussion.

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section 4.Before discussing determinism, though, I’ll explain why I accept the first

premise of the argument. It has become more common lately for substantivalists todeny the first premise; I’ll discuss why I think this is unwise, and why I think muchdiscussion of the first premise is misdirected.

2 Defending The First Premise

Let’s begin with the first premise. What reason is there to believe it? To answer thisquestion I will first discuss the first premise of the hole argument’s predecessor, theLeibniz shift argument. It is easier to first distinguish between good defenses andbad defenses of the first premise of this other argument, and then apply the lessonslearned to the evaluation of the first premise of the hole argument.

2.1 The Leibniz Shift Argument

The first premise of the Leibniz shift argument is this:

(L1) If substantivalism is true, then (assuming that some theory, like Newtoniangravitational theory, that lives in neo-Newtonian spacetime is true) it is phys-ically possible that at each time, each material object be one foot to the leftof where it actually is at that time.3

How might one argue for (L1)? Here’s one way. On one way of formulating New-tonian mechanics, we write down some equations which pick out a set of models.These models are n-tuples of mathematical objects. A typical model may includeR4 and several curves through R4 (functions from R to R4).4 The models represent(or correspond to) physically possible worlds: by looking at the models we mayfigure out what sorts of arrangements of particles in spacetime are permitted by thetheory. But of course by themselves the models tell us nothing about what is phys-ically possible. Only when we also put in place some principles of interpretation,principles which assign representational properties to the models, do they yield in-

3Let’s pretend that ‘to the left’ picks out a determinate direction in space.4These curves must satisfy certain constraints, but they won’t be important.

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formation about physical possibility. Consider, for example, the model that consistsof R4 and just one curve that assigns to each real number r the point (r, 0, 0, 0) inR4. What would the world be like, if this model correctly represented it? We cannottell, until I specify some principles of interpretation. But when I tell you that R4

represents spacetime, that the first component of points in R4 represents temporallocation and the other three represent spatial location, that some of the geometricalrelations between points of R4 represent geometrical relations between points ofspacetime, and that curves in R4 represent the careers of point particles in space-time, then we can see that this model represents a universe in which there is justone point particle moving inertially for all time.

Of course I have not given a complete catalog of the principles of interpreta-tion for these models, and there is room to disagree about just what those principlesare. (Relationalists, for example, may deny that R4 represents spacetime, and mayinstead maintain that it is a fictional device for encoding spatiotemporal relationsamong point particles.) But there is one set of principles which, together with factsabout what models there are, entails (L1). Suppose that in addition to the principlesI gave in the last paragraph we also accept that each point of R4 represents someactual point of spacetime and that it represents the same point of spacetime in everymodel.5 With these interpretive principles in place, consider (again) the model inwhich a particle sits at (t, 0, 0, 0) for all t ∈ R. It follows from facts about the theorythat if there is such a model, then there is also another model in which a particlesits at (t, 1, 0, 0) for all t ∈ R. And since in these two models a particle is located atdifferent points in R4 it follows from what I’ve just said that these models representa particle as located at different points of spacetime. Since each model representsa lone particle moving inertially for all time, these models correspond to distinctpossibilities that differ only with regard to which points of spacetime the particle

5There is an obvious problem with interpreting the models this way, which I willset aside. On this way, if relationalism is true, (and so if there are no actual points ofspacetime), then the substantivalist interpretation of the theory is necessarily false:not only is every model a false representation, no model could have been a correctrepresentation. (Even if there had been points of spacetime, they would not be theactual points of spacetime, and no model can be correct unless the actual pointsexist.)

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occupies. So these possibilities differ merely non-qualitatively. (If we suppose thatthe direction from (0, 0, 0, 0) to (0, 1, 0, 0) is the left-ward direction in space, thenthese models differ only in that according to one of them, the lone particle is shiftedone foot to the left of its location according to the other model, at each time.)

Similar reasoning applies to more complicated models. Assuming that thetheory is true, then, there is a model which represents the world as it is, and anothermodel that represents each material object at each time occupying a region onefoot to the left of the region it actually occupies at that time. Add the premise thatwhatever a model represents is (physically) possibly true, and (L1) follows.

But by itself this is not a good way to argue for (L1). For the mathematicalmodels of our theories don’t come with their representational properties built in.They get their representational properties from us. And substantivalists are not re-quired to give them the representational properties needed to make this argumentwork. A substantivalist could, for example, interpret the models so that the modelsyield only qualitative information about the world. On this second way of interpret-ing the theory, no claims about de re possibility—no claims about which particularpoints of space a particular particle might be located at—follow from inspection ofthe models along with the rules for interpreting them. So, since the first premise ofthe Leibniz shift argument is a claim about de re possibility, it will not follow frominspection of the models along with the rules of interpreting them.

To clarify this second way of interpreting models, return to our two models,one in which a particle sits at (t, 0, 0, 0) for all t ∈ R, the other in which a particlesits at (t, 1, 0, 0) for all t ∈ R. On the second way of interpreting models both ofthese models say the same thing about the world: they both say that there is justone particle moving inertially for all time. Neither says anything about just which(actual) points of spacetime that particle occupies. (Indeed, we cannot even saywhether it is one and the same particle that both models represent.)

So the above argument for (L1) has no force without some argument thatsubstantivalists must give the models one set of representational properties ratherthan another. What might such an argument look like? The most obvious wayto argue that substantivalists should assign models the first set of representationalproperties goes like this: substantivalists believe that it is possible that everything

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be one foot to the left of where it actually is at each time; so they should interpretthe models in such a way that there is a model for each of these possibilities. Butsuch an argument is question-begging in the current context: we are trying to arguefor (L1), so we cannot use (L1) as a premise.

How else might one argue for (L1)? The only real ‘argument’ for (L1) isan appeal to one’s modal intuitions. It seems intuitive to many people that somerestricted combinatorial principle governs possibility.6 In particular, the followingseems intuitive: any way of distributing particles across spacetime is a possible

way of distributing particles across spacetime. That is, as far as (metaphysical)possibility goes, the spatiotemporal location of a given particle is independent ofthe location of any and all other particles. But this principle (together with factsabout what the laws are) entails (L1).

(L1) also seems intuitive when we consider what happens when a substan-tivalist denies it. Denying (L1) is objectionable for the same reason that the moregeneral denial of the existence of possible worlds that differ merely non-qualitativelyis objectionable: it entails implausible essentialist claims.7 (L1) concerns a worldin which everything is shifted one foot to the left at each time. But presumablyanyone who denies (L1) will also deny similar premises asserting that substantival-ists must believe in worlds where everything is shifted two feet to the left, or onefoot to the right, and so on. For short, let’s say that such a person asserts that thereare no shifted worlds. Now, for simplicity, let’s consider the case where there isno time, only (three-dimensional Euclidean) space. Suppose that there are actuallyonly two point-particles, Joe and Moe. Joe and Moe are two feet apart, and theyoccupy points p and q. Then if there are no shifted worlds it is necessary that if Joeand Moe are two feet apart (and nothing else exists), they occupy p and q. And wecan say something stronger. According to (L1) substantivalists must accept possibleworlds that are shifted relative to the actual world. But any reason to believe that

6The unrestricted combinatorial principle is much more controversial. It entailsthat anything could have had any combination of fundamental properties at all. So(given plausible assumptions about which properties are fundamental) it entails thatI could have been a positron, or a point of spacetime, or the successor of four.

7(Adams 1979), (O’Leary-Hawthorne and Cover 1996).

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would be a reason to believe that substantivalists must accept possible worlds thatare shifted relative to any physically possible world. The negation of this claim isthis: for any specification of the distances between n point particles there is exactlyone possible world in which only those n point particles exist, and they have thoseinter-particle distances.8 But how could this be? In the case of Joe and Moe, whatis so special about p and q that makes them the only possible locations of Joe andMoe, when Joe and Moe are two feet apart? Of course there is nothing special aboutthem; if you think that there are no shifted worlds, you must believe that this is justa brute modal fact.9

That’s all I’m going to say to motivate (L1). What further debate there maybe about (L1) can easily be translated into debate about the first premise of thehole argument; since that argument is my focus, I’ll save the further debate for mydiscussion of it.

Since I accept (L1), I will briefly explain where I think the Leibniz shift argu-ment goes wrong. What is supposed to be wrong with recognizing the possibilitiesin (L1)? The possibilities at work in (L1) are qualitatively indiscernible; but what

8I’m continuing to assume that there is no time, only space, and that that space’sexistence is physically necessary.

9Admittedly, there is something special about p and q: Joe and Moe actuallyoccupy them. I find it hard to believe that this explains why they have the modalproperty in question. And there are other brute modal facts that someone whodenies (L1) must accept that cannot be explained in this way. There could havebeen three particles, instead of two, each two feet from the others; if (L1) is falsethen there are three points of space that are the only possible locations of thesethree particles, when they have those inter-particle distances. What makes them sospecial? Not, in this case, that they are actually occupied.

One might claim that there are no shifted worlds and try to avoid these conse-quences by asserting that it is indeterminate where each material object is located.(Presumably it will still be determinately true that there are points p and q such thatnecessarily, if only Joe and Moe exist and they are two feet apart, then they occupyp and q. But there are no points p and q such that it is determinately true of themthat necessarily, if only Joe and Moe exist and are two feet apart, then they occupyp and q.) But facts about where things are located are supposed to be fundamentalfacts; and I don’t believe that there could be indeterminacy where such fundamentalfacts are concerned.

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is bad about qualitatively indiscernible possibilities? Relationalists mention severaldistinct problems. There is an epistemic problem: we can never know which of theindiscernible possibilities is actual. There is a theological problem: God could haveno reason to actualize this world rather than a possibility qualitatively indiscerniblefrom it. The theological problem worries no one these days, and the epistemicproblem is a pseudo-problem.10 What remains is just the bare claim that

(4) There are no qualitatively indiscernible possibilities.

Some find (4) intuitively plausible. I do not, and I also think there are reasons toreject it. Above I argued that substantivalists who accept (4) and deny (L1) mustaccept implausible essentialist claims. As I mentioned above, I believe more gen-erally that anyone who accepts (4), substantivalist or not, must accept implausibleessentialist claims. But I won’t argue for this more general thesis here.

2.2 Back to the Hole Argument

As in the Leibniz shift argument, the first premise of the hole argument asserts thatsubstantivalists must believe in possible worlds qualitatively indiscernible from theactual world. But just what do these alternative possibilities look like? In the con-text of the Leibniz argument, it was easy to say: those were possibilities whichdiffered with regard to where material objects were located. In the context of thehole argument things are not so easy. Now general relativity is our backgroundphysical theory. And general relativity (when given a substantivalist interpretation)differs in ontology from Newtonian mechanics. According to the latter theory (orthe version of it that I had in mind) there was spacetime, on the one hand, with itsgeometrical properties; there were material objects, on the other hand, with their in-trinsic properties (properties like mass and charge); and material objects and pointsof spacetime ‘interacted’ by the former being located at the latter. But general rel-ativity is a field theory. In the mathematical models of the theory, there are vector

10Earman (1989) finds both problems in Leibniz’s letters to Clarke, though Ear-man puts the epistemic problem in verificationist terms. Maudlin (1993) dissolvesthe epistemic problem: it is a priori that each particle is where it actually is at eachtime, rather than shifted one foot to the left.

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and tensor fields (like the metric and the stress-energy tensor) defined on the four-dimensional manifold which represents spacetime. But what is this theory tellingus about the world? Of course, if we’re substantivalists, we think it is telling us thatspacetime exists. But what else is there? Are there in addition a bunch of mathe-matical objects related to the points of spacetime? That is a strange view. Or arethere in addition a bunch of very large material objects—fields—the properties ofwhich vary from point to point? Or is there instead just spacetime, which in addi-tion to its geometric properties, has non-geometric intrinsic properties? (Or is somefourth interpretation correct?)

We can’t settle these questions here. But we need to have some way to char-acterize the possibilities mentioned in the first premise, if we’re going to look at ar-guments that substantivalists need to believe in them. We can of course characterizethe possibilities in very general terms—we can say, they are possibilities which arequalitatively indiscernible, but which differ non-qualitatively because some space-time points ‘play different roles.’ But without further information, we don’t reallyknow what this means.

Let’s suppose we give general relativity the last ontology I mentioned—theone according to which there is just spacetime, with geometric and non-geometricproperties. (Nothing will turn on this choice.) Then the possibility we are asked toconsider is this one: in some future region of spacetime (the ‘hole’), the geomet-ric and non-geometric properties of points of spacetime in that region are ‘pushedaround’ smoothly (so that nearby points stay nearby11), leaving things qualitativelyas they actually are. Of course, putting it this way is slightly misleading, becausewe are ‘pushing around’ geometrical properties. Since the geometry of spacetimeis changing, there is no sense in which the points of spacetime are ‘staying put’while the properties are ‘moving.’ We could just as truly characterize the possibil-ity this way: in some future region of spacetime, the geometric and non-geometricproperties are left where they are, but the points of spacetime are ‘pushed around’underneath them in a suitably smooth manner. And indeed this characterization is

11The ensures that the function we’re using to move the points around is con-tinuous, though actually the function must meet stricter requirements: it must be adiffeomorphism.

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somewhat more accurate. For consider two points inside the hole, p and q, suchthat q ‘gets pushed to where p used to be.’ Now take any point r outside of the hole.r is actually some distance d from p.12 In the possibility we’re contemplating, thedistance from r to p is (probably) not d; instead the distance from r to q is d. Similarfacts hold for other points. So in this non-actual possibility q is playing the ‘geo-metric role’ that p actually plays. (Here is a picture of what is happening: ImagineGod looking down on the spacetime manifold with the properties distributed acrossit as one might look down at the island of Manhattan; imagine Him ‘lifting up’these properties as one might lift up the buildings of Manhattan; imagine Him thenfocusing on some future region of spacetime underneath the properties, and push-ing around the spacetime points in that region as one might then push around thedirt underneath some region of the upper west side (say, the region under ColumbiaUniversity); imagine him finally putting the properties back down, ‘just the waythey were.’ After God’s activity, things are just as they were before, qualitativelyspeaking.)13

2.3 Confusions About The First Premise

What reason is there to believe that the first premise is true? In section 2.1 aboveI discussed a bad argument for the first premise of the Leibniz shift argument. Ananalogous bad argument often appears in discussions of the hole argument. As

12I’m speaking loosely here. Take r to be a point such that there is a unique(either timelike or spacelike) geodesic connecting r and p, and let d be the ‘length’of that geodesic.

13These possibilities are somewhat complicated, because continuum-manypoints are being ‘moved around.’ Are there simpler possibilities that work justas well? Why not take just two spacetimes points in the (absolute) future and havethem ‘switch roles?’ That is, why not just take two spacetime points in the futureand have them exchange all of their geometric and non-geometric properties (in-cluding relational properties like the property of being ten meters from point q)?This possibility is certainly easier to imagine: we simply imagine God focusing hisattention on two spacetime points in the future, reaching down and removing themfrom the spacetime manifold, and then placing each in the hole left by the other.Melia (Melia 1999, section 2.1) argues that these simpler possibilities do work justas well.

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before, this argument appeals to facts about the models (in this case, models ofgeneral relativity) together with contentious principles for interpreting them. Amodel of general relativity is a four-dimensional manifold together with vector andtensor fields defined on it. It is a fact about the theory that if M is a manifold in somemodel then there is another model that also contains M but in which the vector andtensor fields have been pushed around, so that they make the same pattern on M butthe points in M play different roles. (The mathematical device that does the pushingaround is a function from the manifold to itself called a ‘diffeomorphism’ and thetwo models are said to be ‘related by a diffeomorphism.’) We are asked to acceptthe first premise because the theory contains diffeomorphically related models.

So for example Carl Hoefer writes, ‘given the identification [of spacetimewith the manifold; that is, given that the four-dimensional manifolds that appear inthe models represent spacetime], and a straightforward interpretation of the math-

ematical apparatus, the substantivalist is committed to an infinity of qualitativelyindistinguishable possible worlds’ (Hoefer 1996, page 7; italics mine). Hoefersupposes that an interpretation of the models according to which they yield non-qualitative information (an interpretation according to which each point of a mani-fold represents a particular spacetime point, and represent the same spacetime pointin each model in which it occurs) is more straightforward than an interpretationaccording to which models yield only qualitative information. I don’t see how oneinterpretation is any more straightforward than the other. It may be that one fitsbetter with the antecedently established modal commitments substantivalists make;but (as I remarked above) this justification is question begging in the current con-text. Again, arguing for the first premise in this way by looking at the models of thetheory does not work.

Part of the trouble is that the first premise is often stated as a claim aboutthe representational properties of the mathematical models. So Carolyn Brighousewrites, ‘The central claim [of the hole argument] is that the substantivalist has toview diffeomorphically related models as representing distinct situations’ (Brig-house 1994). Similarly, the central question that Jeremy Butterfield thinks the holeargument raises is, Do models related by a hole diffeomorphism represent the samepossible world (1989, page 12)? Stating the first premise in these terms leads to

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confusion. Brighouse’s claim is stronger than the first premise of the hole argumentas I have written it down; it is the conjunction of my premise and some claim aboutthe representational properties of models. So her claim could be false while my firstpremise is true. Arguments against the first premise as Brighouse states it, then, arenot sufficient to block the hole argument.

Although Earman in his book (1989) also presents the first premise as a claimabout models and their representational properties, Earman and Norton (1987) donot seem to make this mistake. They first assert (in the context of the Leibnizshift argument) that substantivalists must accept the possibility of shifted worlds,and then ‘translate’ this claim into a claim about the representational properties ofmodels. Claims about the representational properties of models do not appear aspremises in an argument for the first premise.14

2.4 Defending the First Premise

We still have no argument that substantivalists need to accept the first premise.When I discussed the first premise of the Leibniz shift argument, I gave an argu-ment for it that relied on certain combinatorial intuitions. Can we appeal to thoseintuitions here?

It’s not clear that we have the combinatorial intuitions needed in this case. It isobvious (to me anyway) that (assuming points of spacetime exist) the spatiotempo-ral locations of material objects are independent of each other. But the indiscerniblepossibilities at work in the Leibniz shift argument do not differ with regard to thegeometrical role that each spacetime points plays, as do the possibilities at work inthe hole argument; and it is not nearly so obvious that a point of spacetime couldhave played a different geometric role from the one it actually plays.

One can try to make the two cases look similar, so that the same intuitionsthat support the Leibniz shift also support the hole construction, as follows:

In the context of the Leibniz shift argument, you accept that materialobjects could be located in regions other than the regions at which they

14Though it is true that their paper contains no arguments for the first premise;they seem to think its denial is inconsistent with substantivalism.

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are actually located. But as we are understanding general relativitythere are no things other than points of spacetime. (This is not to saythat according to general relativity people do not exist, or that accord-ing to general relativity there are no tables; they are just not what wethought they were.) So in the context of general relativity the possi-bility according to which you are located in some region other thanthe one in which you are actually located is not correctly described,at a fundamental level, as one in which certain particles bear the lo-cation relation to certain points of spacetime. Instead, it is correctlydescribed as one in which certain non-geometric properties that certainspacetime points instantiate have been changed around in an appropri-ate way. So what your combinatorial intuitions are telling you, in thiscontext, is that there is no impossibility in the idea of changing whichnon-geometric properties certain points of spacetime instantiate. Butthere is no principled distinction to be made, in the context of this the-ory, between the geometric and the non-geometric properties. So youought to admit that it is possible to shuffle the geometric properties aswell.

Why is there no principled distinction to be made? There are several reasons thatcould be offered here. For one thing, just as there are many physically possibleways to distribute non-geometric properties in spacetime, in general relativity thereare many physically possible geometries for spacetime. For another, in generalrelativity different distributions of non-geometric properties in spacetime require asa matter of physical law different distributions of geometric properties.15

This line of reasoning has some plausibility. But it is not nearly as strongas the combinatorial support for the first premise of the Leibniz shift argument.Here’s one reason. The combinatorial intuition I cited was not an intuition thatthe spatiotemporal locations of material objects can be changed around, whateverthe correct physical and metaphysical theory is. Rather, my intuition was this: ifthis is the correct way of describing the world—there are material objects, there is

15Something like this line of thought occurs on page 519 of (Earman and Norton1987).

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spacetime, material objects are located in spacetime—then it seems intuitive thatthe locations of material objects can be changed around.16 So the support for thefirst premise of the Leibniz shift argument does not transfer automatically to thefirst premise of the hole argument.

Still, I accept the first premise. I think it becomes clear that substantival-ists should accept it when we see what is involved in denying it. Denying the firstpremise involves some appeal to essentialism. Since the first premise is about fu-tures in which spacetime points switch roles in the geometry, first-premise deniersoften appeal to some form of geometrical essentialism.17 There are various waysto formulate geometrical essentialism: one could claim that (actual and possible)points of spacetime have their qualitative geometric properties (including their cur-vature properties) essentially; or one could claim that points of spacetime have theirqualitative properties and their non-qualitative relational geometric properties (likebeing ten feet from point p) essentially.

16I think it clear that one’s intuitions about possibility can depend on one beliefsabout fundamental metaphysical matters. Here’s an example. Forget about time fora minute; and suppose space is Euclidean. And suppose it is Euclidean in virtueof the points of space (along with numbers) instantiating a three-place relation, thedistance from x to y is r, in a certain pattern. Then it seems perfectly possiblethat every pair of points of space be twice as far apart as they actually are. Forthis simply involves each pair m and n instantiating the distance from x to y is2r iff they actually instantiate the distance from x to y is r, where r is some realnumber. But now suppose instead that space is Euclidean in virtue of the points ofspace instantiating two relations, betweenness and congruence, in a certain pattern.Then it seems impossible that every pair of points of space be twice as far apartas they actually are—because there are no absolute facts about how far apart twothings are, there are only comparative facts about whether two things are the samedistance apart as two other things. (One might assert that ‘the congruence relationcan hold between points of space in different possible worlds,’ and that this allowsone to make sense of the possibility in question; but this assertion only makes senseif one is a modal realist, which I am not. And even modal realists will hesitateto say that the congruence relation can hold between points of space in differentpossible worlds: Lewis defines ‘possible world’ as ‘maximally spatiotemporallyrelated concrete object.’ It follows from this definition that parts of distinct possibleworlds bear no spatial relations (and congruence is a spatial relation) to each other.)

17(Maudlin 1990).

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We should reject these versions of geometrical essentialism. In general rela-tivity, the geometry of spacetime depends on the distribution of mass-energy.18 So,if general relativity is true, then if I had raised my hand a moment ago, the geometryof the region of spacetime around me would have been different. So, if these ver-sions of geometrical essentialism are true, then if I had raised my hand a momentago, (part of) the region of spacetime I actually occupy would not have existed. Butcertainly it was up to me whether I raised my hand a moment ago; so if these ver-sions of geometrical essentialism is true, it was up to me whether a certain regionof spacetime exists. But that is absurd.19

There is a weaker version of essentialism that does not entail that what pointsof spacetime exist depends on what I do. It is an instance of the more generaldoctrine that there are no possible worlds that differ merely non-qualitatively. Onthis view it is impossible that (a) all the actual points of spacetime exist, (b) thegeometry of and distribution of matter in spacetime is just as it actually is, and (c)some points of spacetime play different roles than they actually play. But this viewdoes not entail that a given spacetime point have any particular curvature propertyessentially, or that it must be any particular distance from some other point. So thisview evades the objection above: it is consistent with this view that even if I hadraised my hand, all the actual points of spacetime would still have existed.

What motivates those who appeal to geometrical essentialism is the idea thata point of spacetime cannot be separated from its geometrical properties. This idea

18The argument to follow depends on special features of general relativity; ana-logues of the hole argument in the context of other physical theories are immune toit.

19Maudlin replies to an argument like this one by ‘appealing to counterpart the-ory’ (1990, page 550). That is, after arguing that ‘spacetime (the actual spacetime)could have had a different geometry’ is false, he proposes a counterpart-theoreticsemantics for our modal vocabulary on which ‘spacetime (the actual spacetime)could have had a different geometry’ is true. But for this response to work, he mustclaim that my objection seems plausible when we understand the modal vocabularythat occurs in it in the new way, but does not seem plausible when we understandit in the old way. But I don’t think this is so; when I wrote the objection down,I meant to be using the modal vocabulary in just the way Maudlin does when hedefends essentialism; and the argument seems plausible when read that way.

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does not motivate the weaker essentialism I am now discussing. I gave my reasonsfor rejecting its motivation, the doctrine that there are no merely non-qualitativedifferences, above on page 32: it leads to implausible essentialist claims. So I lookelsewhere for a response to the hole argument.

2.5 Counterpart Theory

Butterfield asserts that one can deny the first premise of the hole argument withoutbeing an essentialist by appealing to counterpart theory (1989, page 22). How mightthis work? Essentialism is a collection of de re modal claims. Counterpart theory isnot; it is (part of) an analysis of de re modal claims. (Roughly speaking, de re modalclaims like ‘Nader could have won’ are analyzed not as ‘There is a possible worldin which Nader wins,’ but as ‘There is a possible world in which a counterpart ofNader (someone sufficiently similar to him) wins.’) And counterpart theory is notincompatible with essentialism. There are counterpart relations that make certainessentialist claims true: there is a counterpart relation, for example, which makestrue the de re modal sentence, ‘I could not have been a poached egg.’

Of course there are other counterpart relations that make this and many otheressentialist claims false. To use counterpart theory to deny the first premise withoutembracing essentialism, then, one must find one of these anti-essentialist counter-part relations that makes the first premise false. But this cannot be done. There areno such counterpart relations because the denial of the hole argument’s first premiseis equivalent to a version of essentialism. The first premise just is the claim that cer-tain spacetime points can switch their geometrical roles; for this premise to be falseis for the points to have (some aspect of) their geometrical roles essentially. Thisis so whether you analyze de re modal claims in terms of counterparts or not.20

20In Lewis’s original version of counterpart theory, as presented in ‘CounterpartTheory and Quantified Modal Logic’ (1968), he made it an axiom that each thinghas at its own world only one counterpart: itself. This seems to lead immediately tothe result people like Butterfield want: for (reverting for the moment to the Leibnizshift argument) you might think that the truth of ‘Each thing could have been onefoot to the left of where it actually is’ requires each point of space to have a coun-terpart at its own world other than itself. This is still essentialism: but one mightthink that it is made more palatable by following immediately from Lewis’s theory.

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This argument shows more than that appealing to counterpart theory is not a wayto avoid essentialism while denying the first premise. It shows that the only way todeny the first premise is to embrace some form of essentialism.

3 The Second Premise

I have said that substantivalists should accept the first premise of the hole argument,and so accept that (if general relativity is true) there is more than one physicallypossible future consistent with the way things are now: the actual future, and afuture that is qualitatively indiscernible in which some future spacetime points haveswitched roles. Now, the second premise of the hole argument is:

(2) If there is more than one physically possible future consistent with the waythings are now, then determinism is false.

To uphold the claim that general relativity is a deterministic theory, then, substan-tivalists should deny this second premise. I know this premise looks analytic; butthis is merely an appearance. There are reasons to reject it which are independentof its role in the hole argument.

3.1 Determinism: Examples and Counterexamples

It is easy to convince yourself that not just any alternative physically possible futurecounts against determinism, as we ordinarily think of it. If one accepts that a givenworld has futures that differ merely non-qualitatively, then it should be intuitive thatthose futures do not count against determinism.21

But, in the first place, it does not follow immediately: we need an extraassumption—namely that there are no qualitatively indiscernible worlds—whichLewis does not make. And, in the second place, Lewis himself came to think thisaxiom too restrictive and abandoned it in On The Plurality of Worlds (Lewis 1986b).

21David Lewis’s analysis of determinism in (Lewis 1983) entails that alternativefutures that differ merely non-qualitatively do not count against determinism. Heoffers no explanation of why this should be, though. Butterfield’s analysis in (But-terfield 1989) and Brighouse’s in (Brighouse 1997) are equivalent to Lewis’s. I lookat Lewis’s analysis in more detail below.

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Joseph Melia (Melia 1999) gives an example of a world with possible fu-tures that differ merely non-qualitatively, but which seems, intuitively, determinis-tic. Consider these simple laws: nothing ever moves; there are two kinds of parti-cles, P+ and P− particles; each P+ particle decays into a P− particle five minutesafter coming into existence; the P− particles occupy the same places as the P+ par-ticles that give birth to them. P− particles never decay. (We are also to assumethat worlds governed by these laws are relationalist: there is no space or time, onlyspatiotemporal relations between particles).22 Suppose that the history of the worldup to a certain time looks like this: in the beginning there was nothing, and thenthree minutes ago two P+ particles were created three meters apart. It appears thatthe future is determined: in two minutes both P+ particles decay into P− particles.There is no other possibility. But if we are working with the conception of deter-minism at work in the second premise of the hole argument, then this world comesout indeterministic. Call one of the P+ particles α+, the other β+. Then α+ actuallydecays into α−, β+ into β−. But α− and β− could have switched roles. So there aretwo possible futures (differing merely non-qualitatively) for the world; they differwith regard to which P− particles α+ and β+ decay into. But it is absurd to allowthese differences count, when asking whether this world is deterministic.

If this line of thought is right, then we have reason to deny the second premise.Unfortunately there is an example which seems to show that this line of thought isnot right, that we do allow the existence of physically possible futures which differmerely non-qualitatively to count against determinism.23 Exert enough downwardforce on a cylindrical column and it will usually collapse into an elbow shape. Weare to imagine a non-actual set of laws of nature according to which cylindricalcolumns, when they collapse, always collapse into an elbow shape, even when thesituation is perfectly symmetric around the axis of the cylinder—the force is appliedstraight down on the center of the top of the cylinder, the cylinder itself is not

22But wait: then what do we mean when we say ‘the P− particles occupy thesame places as the P+ particles that give birth to them’? Perhaps Melia means toallow for cross-time distance comparisons, so we can say that some P− particle iszero meters from the P+ particle from which it decayed.

23The following example first appeared in (Wilson 1993).

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weaker on one side, and so on. (We are also to imagine that worlds governed bythese laws contain Absolute (Newtonian) space and time.) Now imagine a worldcontaining such a cylinder in which a sufficient force is applied straight down onthe center of the top of the cylinder, causing it to collapse, but which is perfectlysymmetric before the cylinder collapses. (The world contains only the cylindersitting on a perfectly spherical planet and a sphere falling directly on top of thecylinder.) Intuitively, this world is not deterministic: the laws do not dictate inwhich direction the cylinder will collapse. But the different possible futures—thecylinder collapses this way, or it collapses that way—are qualitatively indiscernible.

What does this example show? Even if we agree that the column world isindeterministic, it does not show that the second premise is true. For I, at least, stillhave the intuitions that in some cases qualitatively indiscernible futures do not countagainst determinism: unless more is said, the collapsing column example does notsuggest that these intuitions were misleading.

So we are left with the task of finding an analysis of ‘determinism’ whichsometimes allows merely non-qualitative differences to count against determinism—as in the column world—and sometimes does not—as in the worlds in Melia’s ex-ample. And we want an analysis that is appealing for independent reasons, not justbecause it allows us to defend substantivalism from the threat of the hole argument.

3.2 Analyzing Determinism

There is such an analysis available. Consider the Laplacian picture of determinism:in a deterministic world, an extraordinarily intelligent demon who knew all the in-formation about the present as well as the laws of nature would be able to deduceall the information about other times.24 But what do we mean, all the information?We could mean, all the qualitative information; we could mean all the qualitativeand non-qualitative information; or we could mean something in between. So this

24This picture speaks of agreement at a time forcing agreement at all times. Thereare other varieties of determinism, differing with regard to which regions of space-time worlds must agree on to force agreement through spacetime (Earman (1986)contains a survey). But this is the one in play in discussions of the hole argument,so it is this one I shall focus on.

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gives us several ways to make the Laplacian picture more precise: we can be moreprecise about what information we give the demon to start with, and more preciseabout what information we demand that he produce. Here’s one way to make thesemore precise: we could give the demon all the qualitative information about thepresent, as well as the laws of nature, and demand that he produce all the qualita-tive information about other times. Here’s another: we could give the demon all thequalitative and non-qualitative information about the present, as well as the laws ofnature, and demand that he produce all the qualitative and non-qualitative informa-tion about other times. (This is the conception of determinism relationalists rely onin the hole argument.) The analysis I endorse captures the only natural middle po-sition. We give the demon all the qualitative and non-qualitative information aboutthe present, as well as the laws of nature. (If we were to write down the informationhe is given in a language the predicates of which express only qualitative properties,some of the sentences we would write down would contain proper names of thingsthat exist in the present.) And we demand that he produce all the qualitative infor-mation about other times, and also all the non-qualitative information about othertimes which (if we were to write it down) can be expressed using only the propernames we have already given him. (So he need not produce any non-qualitativeinformation that can only be expressed using proper names of things that exist onlyin the future.) If he cannot do this, then the world is not deterministic.

This picture of determinism meshes with our intuitions about determinism.Consider Melia’s particle decay world. No Laplacian demon is needed to writedown the relevant information; we can do it ourselves: given that three minutesago two P+ particles, α+ and β+, were created three meters apart, we know that intwo minutes there will be two P− particles, one having decayed from α+, the otherfrom β+, and they will be three meters apart, for the rest of time. This world isdeterministic, even if we cannot deduce which P− particle decays from which P+

particle; even if an extraordinarily intelligent demon cannot, either. For in order toexpress the required proposition we must use names for the P− particles; and we arenot required to deduce information that can only be expressed using such names.25

25By ‘names’ I mean names that do not have their reference fixed using descrip-tions. We can, of course, deduce that α+ decays into α−, because I introduced ‘α−’

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The column world is indeterministic, though, because (since the points of spaceendure through time and exist at the initial time) we must be able to deduce all thenon-qualitative information expressible using the names of the points of space. Sofor any point p we must be able to deduce whether the column will collapse towardpoint p or not; and this we cannot do.26

This picture of determinism also seems intuitively correct, when consideredon its own. I have said that we should demand that the demon produce only thatnon-qualitative information about other times which can be expressed using onlythe proper names we have already given him. And if we are going to demand thatthe demon produce some non-qualitative information, how could we reasonably de-mand more information than this? Think of proofs from some given set of premisesin some formulation of the first-order predicate calculus. It is impossible to de-rive any sentence containing names which do not occur in the premises you aregiven.27 How can we ask of the demon something that, as a matter of logic, cannotbe done?28

by saying it names the particle into which α+ decays.26The use of Absolute space is not necessary; even if the history of the world

unfolds in neo-Newtonian spacetime, we have names for currently existing pointsof spacetime, so we must be able to deduce whether the column will collapse towardthe unique spacetime point that exists at the time of collapse and is co-located withp relative to such-and-such frame of reference. This, too, we cannot do.

27Well, not exactly. But it is true that for any derivable sentence pϕ(a)q contain-ing a constant a not occurring in the premises, you can also derive p∀xϕ(x)q; so thenon-qualitative information expressible using new names that you can derive is notsubstantive.

28In this argument, I assume that the laws of nature are purely qualitative; theydo not, as it were, mention any particular individuals by name. There are theoriesof laws of nature which may allow non-qualitative laws. The best system theory oflaws (Lewis 1983), in particular, may do this: in simple enough or strange enoughworlds, the strongest and simplest system may be one containing theorems whichmention some particular individual by name. I think my analysis could be amendedto accommodate such laws.

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3.3 A Formal Definition

I began with the Laplacian picture of determinism and distinguished three ways ofmaking it more precise. The three ways agree that a deterministic world is one inwhich a powerful demon, when given complete qualitative information about onetime and the laws of nature, can produce complete qualitative information about allother times. They disagreed over how much non-qualitative information we givethe demon about the initial time, and how much non-qualitative information wedemand he produce about other times. I favored the precisification according towhich we give the demon all non-qualitative information about things that exist atthe initial time, and demand that he produce all non-qualitative information about

just those things at all other times.But this talk of non-qualitative information and of what the demon can pro-

duce is still vague. I will now precisify it further. To see how, first consider howto precisify the version of Laplacian determinism that deals only with qualitativeinformation. To give the demon all the qualitative information about a time is to tellhim how many things exist at that time, which qualitative properties each of theminstantiate, and for each n (where n might be infinite), which qualitative n-place re-lations any n-tuples of them instantiate. A world that is deterministic in this (purelyqualitative) sense, then, is one in which the demon, given all qualitative informationabout a time and knowing the laws of nature, can tell us how many things there arein total, what qualitative properties each of them instantiate, and so on. And therewill be some time slice of the world he describes that will match the initial time wedescribe to him, where ‘match’ here means ‘be a qualitative duplicate of.’ Althoughthe demon can tell us how many things exist in this world, though, he cannot tell uswhich things exist, and cannot distinguish between worlds in which two things haveswitched roles. Dispensing with the demon, this means that a world w is determin-istic (in this purely qualitative sense) just in case any other world that is physicallypossible relative to w and is a qualitative duplicate of w at a time is a qualitativeduplicate of w full-stop.

We have almost arrived at David Lewis’s analysis of determinism. Lewis’sanalysis makes use of an analysis of qualitative duplication; and this analysis itself

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makes use of the distinction between properties that are perfectly natural and thosethat are not. The perfectly natural properties (and relations) are the fundamentalproperties (and relations); wherever any other property or relation is instantiated,it is instantiated in virtue of the global pattern of instantiation of the fundamentalproperties and relations. The set of perfectly natural properties and relations, then,forms a supervenience base for the set of all properties. (I will also assume thatthe perfectly natural properties are purely qualitative.) Two things are duplicates,then, iff they share all the same perfectly natural properties, and their parts can beput into correspondence so that corresponding parts share the same perfectly naturalproperties, and corresponding pairs of parts stand in the same perfectly natural (two-place) relations (and so on). Lewis’s analysis of determinism, then, is

(D0) A possible world w is deterministic iff every world that is a duplicate of w ata time and is physically possible relative to w is also a global duplicate of w.

The column world is a counterexample to Lewis’s analysis. We want a Lewis-styleanalysis of determinism that corresponds to my modified version of the Laplacianpicture in the way that Lewis’s corresponds to the version that takes only qualitativeinformation into account.

To give the demon all the qualitative information about a time and all thenon-qualitative information about things that exist at that time is to tell him not justhow many things exist at that time, but also which things exist at that time, and whatqualitative properties each instantiates at that time (and so on). So the demon candistinguish between times that are qualitative duplicates in which some things haveswitched roles.

A deterministic world, then, is one in which the demon, given the relevantinformation about a time and knowing the laws of nature, can tell us not just howmany things there are in total and what qualitative properties each instantiates (andso on), but can also tell us which of them are the things that exist at the initialtime. So the demon can tell us what roles the things that exist at the initial timeare playing in the global structure of the world. And there will be some time sliceof the world he describes that will match the initial time we described to him; andmatch not just because it is a qualitative duplicate, but also because the same thingsexist and play just the same roles.

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To express this as a Lewis-style definition I need the concept of a duplicationfunction. Recall that two things are duplicates iff their parts can be put into a cor-respondence meeting certain conditions. Call such a correspondence a ‘duplicationfunction.’ (Two things can be duplicates according to more than one duplicationfunction: think of two congruent equilateral triangles.) The analysis is:

(D1) A possible world w is deterministic iff every world that is a duplicate of w ata time and is physically possible relative to w is also a global duplicate of w,under a duplication function that is the identity function on the initial times.29

Worlds at which general relativity is true are deterministic on this analysis, as theywere on Lewis’s. (The diffeomorphism that generates the hole is a duplication func-tion; since it only changes which spacetime points play which roles in the future,it is the identity on the initial times). But the collapsing column world is indeter-ministic. Again, let ‘p’ and ‘q’ name points of space that lie in different directionsfrom the column. There is a world w in which the column collapses toward p and

29I’m assuming that we can speak of two things existing in more than one pos-sible world. Counterpart theorists might be nervous about such talk of transworldidentity appearing in an analysis of ‘determinism,’ but the analysis can be easilyre-written in terms of counterpart relations: say that

w is deterministic iff for any world w′ that is physically possible relative to wand duplication function d, if a time slice of w and a time slice of w′ are dupli-cates under d, then w and w′ are duplicates under some duplication functiond∗ that agrees with d on the initial time slices.

Here the function d is a counterpart relation between the two time slices. It providesthe standard for identifying things that exist on the time slice in w with things thatexist on the time slice in w′.

(D1) first appeared in (Belot 1995), though he formulates it in counterpart-theoretic terms, as does Melia (1999). Belot does not endorse (D1), but Melia does.Melia does (and Belot does not) attempt to show that there is something natural andintuitive behind this formal analysis, that it is not just an ad hoc device for avoidingthe conclusion of the hole argument. But Melia’s intuitive characterization is dif-ferent from mine. He appeals to branching possible worlds. But at its best this willonly allow us to explain senses of determinism in which the past and the laws fixthe future. My characterization can be generalized to other senses of determinism.

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a world w∗ in which it collapses toward q.30 So a duplication function between w

and w∗ must map p to q. But p and q are distinct and exist at t, before the towercollapses. So no duplication function from w to w∗ can act as the identity on t.

4 Earman’s and Stein’s Argument

In Chapter 3 of World Enough and Space-Time Earman (following Stein (1977))argues that relationalism about motion, together with the possibility of determinism,entails relationalism about ontology.

Pause to consider how implausible this is. Suppose relationalism about mo-tion is true, and so that the world contains a Machian spacetime. (The discussion tofollow is unchanged if we consider Leibnizian spacetime instead.) Suppose that theone law governing particle behavior is this: inter-particle distances never change.The law looks deterministic: if I know there are just two particles, and that they aretwo feet apart right now, I know all there is to know about the future: there willcontinue to be two particles, two feet apart. Of course I won’t know if they’re inthe same place later as they are now, or whether they’re in the same place relative tosome frame of reference as they are now. But I couldn’t know these things, becausein Machian spacetime there is no sense to be made of ‘same place across time,’ evenrelative to some frame of reference.

Earman’s argument appeals to

(SP2) Any spacetime symmetry of theory T is a dynamical symmetry ofT .

Rotations are symmetries of Euclidean space; by performing a (possibly different)arbitrary rotation on each instantaneous Euclidean space in Machian spacetime weobtain a symmetry Φ of that spacetime. If we suppose that spacetime is populated

30There is a debate in modal metaphysics over whether distinct but qualitativelyindiscernible possibilities really require distinct possible worlds. (Some defendersof counterpart theory, like David Lewis (1986b), say no.) This makes no differenceto my argument; my argument works even if w and w∗ are the same world, thoughin some cases I would have to phrase the argument in terms of counterpart theory(see footnote 29 above).

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by particles, then this symmetry is a dynamical symmetry of theory T just in casefor any model M there is another model MΦ with the same spacetime manifold suchthat a point x in the spacetime of M is occupied by a particle iff Φ(x) is occupied inthe spacetime of MΦ. In his argument Earman presupposes (what I have complainedabout earlier) that substantivalists must say that distinct models of T correspond todistinct possible worlds.

The argument goes like this. Suppose relationalism about motion and sub-stantivalism are both true. Then spacetime has a Machian structure. Suppose thereare some particles obeying some laws. The function Ψ on Machian spacetime whichis the identity for all times before and including t but rotates each of the later instan-taneous Euclidean spaces through some angle about some axis after t is a symmetryof Machian spacetime. By (SP2), Ψ is also a dynamical symmetry. So the stateof the world at t plus the laws fails to fix the state of the world after t: they fail todetermine whether the particles are where they actually are after t, or are where theywould be if rotated through some angle about some axis. That is, if relationalismabout motion and substantivalism are both true, then no theory of particle motionis deterministic. If relationalists about motion want to allow for the possibility ofdeterminism, they must also be relationalists about ontology.

These two futures at work in Earman’s argument differ merely non-qualitatively:they differ merely over which spacetime points in the future the particles occupy.Earman’s argument presupposes that the existence of futures that differ in this wayis enough to render a theory indeterministic. As I have argued, this is not so. Ψ isa global duplication function that acts as the identity on the original times; so thedistinct futures generated by it do not render the theory indeterministic, accordingto the analysis of determinism I have defended. Relationalism about motion doesnot require relationalism about ontology; and good thing, too, since I argued theopposite in chapter 1.

5 Conclusion

I have argued that the hole argument fails because its second premise is false. Thatpremise may seem true at first, but only when read with an incorrect analysis of

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‘determinism.’ I have articulated and defended a better analysis of ‘determinism,’one that is a natural precisification of the Laplacian conception of determinism, thathas intuitive appeal considered on its own, and that fits our intuitions about severalcases. And I have shown how this modified Laplacian conception can be givenformal expression in terms of possible worlds.

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Chapter 3

Supersubstantivalism

The best current thinking does not claim that particles arenot built out of spacetime....For the time being, as a meansto get on with the world’s work, and to deal with particles ona practical working basis, it makes sense to treat particles asif they are foreign objects. This working procedure does notexclude any longer-term possibility to account for a particlein terms of geometry—as one today accounts for the eye ofa hurricane in terms of aerodynamics, and the throat of awhirlpool in terms of hydrodynamics. (Wheeler and Taylor1963, page 193)

1 Introduction

Substantivalists believe, and relationalists deny, that spacetime exists. Set relation-alism aside; substantivalists still have plenty to disagree about.

There is room for them to disagree about the nature of material objects. Somesubstantivalists—call them ‘dualists’—hold that material objects are distinct fromspacetime. Others—‘supersubstantivalists’, as Sklar calls them1—hold that mate-rial objects (if there are any) are identical with regions of spacetime. Supersub-

1In Space, Time, and Spacetime (Sklar 1974). I believe that Sklar coined thisterm; it first appears on page 214.

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stantivalists identify material objects with the regions of spacetime that dualists saythose material objects occupy. Dualists, then, like their namesakes in the philoso-phy of mind, accept additional ‘ontological categories’: just as mind-body dualistshold that minds differ in kind from physical things, so dualists in this debate holdthat physical objects differ in kind from points and regions of spacetime. As JohnWheeler puts it, dualists regard material objects as ‘strange and nongeometricalobjects immersed in spacetime’ (Wheeler and Taylor 1963, page 191).

Supersubstantivalism may initially seem incredible, but that has not stoppedsome very smart people from believing it. Descartes identified body with exten-sion, and so looks like a supersubstantivalist. Bennett (1984) attributes super-substantivalism to Spinoza. More recently, Ted Sider (2001) briefly defends super-substantivalism in the course of arguing for the doctrine of temporal parts.2 Andsupersubstantivalism is not just a metaphysician’s fantasy. Respected physicistshave held views close to or identical with it: Isaac Newton was a supersubstantival-ist, and so was Einstein during one period of his career. And the many physicistswho believe that there are only fields (like the 19th century physicists who iden-tified charged particles with parts of the electromagnetic field3) are close to beingsupersubstantivalists. This is an impressive catalog.

There are several varieties of supersubstantivalism. In the next section I’llsurvey these varieties, and then I’ll look at arguments for and against the view.

2 Varieties of Supersubstantivalism

Dualism is straightforward: there is a special fundamental relation, the occupationrelation. Every concrete material object bears the occupation relation to some re-gion of spacetime.

Generic supersubstantivalism is also straightforward: there is no such funda-mental relation as the occupation relation. Every concrete material object is identi-cal with some region of spacetime. But there are more and less radical varieties ofsupersubstantivalism.

2Other philosophers who flirt with supersubstantivalism include David Lewis(1986b), Hartry Field (1989), and W. V. Quine (1981).

3The ‘electromagnetic view of nature’ is discussed in (McCormmach 1970).

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Different versions of supersubstantivalism give different accounts of whatfundamental properties and relations points and regions of spacetime instantiate. Atone extreme lies the view that the only fundamental properties and relations space-time instantiates are geometrical ones. Hard-headed scientifically-minded philoso-phers are likely to respect this version most, because it is linked to a scientificresearch program in a way that the others are not. (Sklar considers this version aworthy topic of discussion, while dismissing a more modest version as a ‘linguistictrick’ (1974, pages 166, 233).) Inspired in part by the general theory of relativity,John Wheeler advocated this variety of supersubstantivalism. In general relativityEinstein eliminated the gravitational force: the motions that this force explainedare in Einstein’s theory explained instead by the curvature of spacetime. Wheelerhoped that all forces could be eliminated, and all motion explained by the curva-ture of spacetime. (It’s this hope that he expresses in the quotation that heads thispaper.) Of course, the sense in which such a theory ‘explains motion’ is now a lit-tle funny: if we’re supersubstantivalists, we don’t think there are particles distinctfrom spacetime, particles that move by being located in different places at differenttimes, whose motions we can attempt to explain. When an extreme supersubstan-tivalist claims to explain the motion of bodies he means something like this: thewhole story of the history of the universe is contained in the geometrical structureof spacetime; there are laws governing that structure which allow us to explain whya given instantaneous space has the geometry it does by citing the geometry of ear-lier instantaneous spaces together with the laws; and ordinary talk of the motion ofbodies is in some sense talk of the curvature of spacetime.4 The Grand Unified The-ory that Einstein sought was to be a theory that provided explanations like this: thetheory would not postulate (dualistically-conceived) particles, but certain regionsof spacetime in possible worlds permitted by the theory would look ‘particle-like’(see (Earman 1995, page 16)). Producing such explanations is a real challenge, one

4Not all spacetimes permitted by general relativity are well-behaved; some can-not be divided up into a series of instantaneous spaces ordered in time. In suchspacetimes we may not be able to explain why a given spacelike region of space-time has the structure it does by appealing to the structure of some earlier spacelikeregion and the laws.

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that has not yet been met: while Einstein showed how to ‘reduce’ gravity to cur-vature, no one has succeeded in reducing electromagnetism, or the other forces, tocurvature in a plausible way.

Although the success of general relativity helped motivate this research pro-gram, its roots go back farther. Back in 1870 W. K. Clifford, inspired by Riemann’stheory of curved surfaces, hypothesized that what we ordinarily call the motion ofmatter is nothing but ‘variation of the curvature of space,’ and that ‘in the physicalworld nothing else takes place but this variation.’ Like Wheeler, Clifford hopedto produce scientific explanations of phenomena that appeal only to the geometryof space: he hints that ‘I am endeavoring in a general way to explain the laws ofdouble refraction on this hypothesis’ (1882, page 22). Even earlier Newton secretlytoyed with a version of supersubstantivalism according to which the only funda-mental properties space instantiates are geometrical, though he makes no attempt touse the curvature of space to explain motion. (Newton thought space was flat, andin his time the geometry needed for dealing with surfaces with variable curvature—Riemannian geometry—had not been invented. I’ll look at Newton’s theory below.)

I’ve been discussing extreme versions of supersubstantivalism. Other ver-sions are more modest. More modest versions are more liberal about what funda-mental properties they allow spacetime to instantiate. One modest version acceptsas fundamental the (non-geometrical) fundamental properties we ordinarily thinkthat fundamental particles instantiate—properties like mass and charge—and al-lows spacetime to instantiate them.

This more modest version does not face the daunting challenge that the ex-treme version does: that of rewriting the laws of electromagnetism and the otherforces as laws governing the curvature of spacetime. In fact, existing spacetime the-ories (like Newtonian gravitational theory and electromagnetism) can easily be re-interpreted to make them compatible with the more modest form of supersubstantivalism.

3 Is Dualism the Default View?

Strictly speaking, dualism and supersubstantivalism are not incompatible. Both aretrue in a world with spacetime but no material objects. But our world is not one of

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these worlds, so we must choose.Dualism seems to most substantivalists like the default position. They want

powerful arguments showing that dualism is incoherent, or that it conflicts withsome other strongly held beliefs, before they take supersubstantivalism seriously.I don’t know of any arguments like that; so if that is what is required, super-substantivalism is in bad shape.

How does dualism get this default position? Not because we can just see

that material objects aren’t regions of spacetime. Maybe substantivalists give du-alism default status because it is entrenched: maybe substantivalists have believedit for a long time, are used to picturing the world in its terms, and have never re-ally considered the alternative. But psychological entrenchment does not justifygiving dualism default status.5 If preferring supersubstantivalism requires strongarguments against dualism, that must be because there are good reasons in favor ofdualism in the first place. But what are these reasons?

I can think of two reasons we give dualism default status, and substantivalistsshould not think either are any good.

First, we might oppose supersubstantivalism because we feel some motivationfor relationalism. I’m sitting in my chair and my bamboo plant is across the room,and (it seems) there is absolutely nothing between us. But if supersubstantivalismis correct then we are both swimming in a sea of regions of spacetime, our bor-ders contiguous with other things made of the same stuff that we are. So super-substantivalism conflicts with the way things ordinarily seem.

Even if this is a correct account of the way things ordinarily seem—andsurely sometimes it is—substantivalism already conflicts with it. And I do notsee that dualism comes closer to matching the ordinary appearances than super-substantivalism.

Second, we might oppose supersubstantivalism because it sounds odd to saythat I bought my coffee from a region of spacetime. But, in general, the demiseof ordinary language philosophy has taught us to be suspicious of arguments thatstart with ‘it would sound odd to say....’ And in particular, history should make us

5Maybe those who accept ‘conservative’ accounts of belief revision (like Har-man (1986)) will disagree.

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suspicious: there was a time when it sounded odd to say that I bought my coffeefrom a swarm of elementary particles.6

There are, in addition, more sophisticated philosophical reasons substanti-valists might oppose supersubstantivalism. Maybe supersubstantivalism is falsebecause material objects move but regions of spacetime do not; maybe it is falsebecause material objects lack but regions of spacetime have temporal parts; maybeit is false because my bamboo plant could have been located elsewhere, but noregion of spacetime could have been located elsewhere. These are important objec-tions, and I address them below. But I don’t think they capture the reasons why weinitially recoil from supersubstantivalism and give dualism default status.

I conclude that substantivalists should not give dualism default status. Weshould consider dualism and supersubstantivalism as competitors on an equal foot-ing, and see where argument leads.

So what reasons are there to be supersubstantivalists? In sections 4 through 7I discuss instances of two strategies for defending supersubstantivalism, one due toDescartes, and the other due to Newton.

4 Cartesian Supersubstantivalism

Descartes is typically classified as a relationalist. But while Descartes was cer-tainly a relationalist about motion, it is less clear that he was a relationalist aboutontology, rather than a supersubstantivalist. (I argued in previous chapters that re-lationalism about motion is not only consistent with substantivalism, it requires it.)In the Principles of Philosophy Descartes writes, ‘in reality the extension in length,breadth and depth which constitutes a space is exactly the same as that which con-stitutes a body’ (2:10).7 That sounds like a strong (and not particularly plausible)version of supersubstantivalism: not only is every material object (or ‘body’) iden-tical with some region of space, but every region of space is a material object.(Since Descartes lived in a pre-relativistic age, this version of supersubstantivalismidentifies material objects with regions of space, not with regions of spacetime.)

6(Sider 2001, sect. 4.8).7Unless otherwise noted, all citations of Descartes are to the Principles

(part:section).

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Is Descartes really a supersubstantivalist? Field (1989) claims that supersub-stantivalists can be distinguished from relationalists by asking the following ques-tion: is it possible that there be empty regions of spacetime? Only supersubstanti-valists will answer ‘yes.’ Descartes says ‘no,’ and so counts as a relationalist by thiscriterion. But I’m not sure this criterion is a good one. A dualist might think that itis necessary that every region of spacetime is occupied by a material object. Surelya supersubstantivalist can accept the analogous claim, that it is necessary that everyregion of spacetime is a material object. Still, it may be that there is a merely se-mantic difference between some relationalists and supersubstantivalists—they justdiffer over how they are willing to use the words ‘region of space(time).’ I will setthis issue aside and speak as if Descartes is properly classified as a supersubstanti-valist.

Descartes’s argument for Cartesian supersubstantivalism rests on the claimthat ‘the nature of body consists in...its being something which is extended in length,breadth and depth’ (2:4). If we say that talk of something’s nature is talk of theconjunction of all of its essential properties (and this does seem to be what Descarteshas in mind in 2:11), then Descartes’s claim amounts to this:

(1) Being spatially extended is the only essential property of material objects.

Descartes also accepts a partial converse of (1), namely

(2) Necessarily, every extended thing is a material body.

But the arguments he gives in 2:4 and 2:11 (which I examine below) are clearlyarguments for (1), and without extra premises (1) does not entail (2). Nor is (2)sufficient to establish supersubstantivalism: it entails that every region of space is amaterial body, but not that every material body is a region of space.

In 2:11 Descartes says ‘Yet this [namely, the idea of something that is ex-tended in length, breadth, and depth] is just what is comprised in the idea of space.’I take this to mean

(3) Being spatially extended is the only essential property of regions of space.

This looks plausible. And we can get from (1) and (3) to Cartesian supersubstantivalismif we add another premise:

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(4) Distinct kinds of things cannot share all of their essential properties.

So there is only one kind of thing that has spatial extension as its only essentialproperty; and we have two names that apply to (all and only) things of this kind:‘region of space’ and ‘material object.’8

This argument aims to establish a particular version of supersubstantivalism.So we can evaluate it from two points of view. First, does it give us reason to acceptsupersubstantivalism in the first place? Second, if we are already supersubstantival-ists, does it give us reason to accept Cartesian supersubstantivalism instead of someother version? I will take these questions in reverse order.

Cartesian supersubstantivalism is hardly the most plausible version of super-substantivalism. Supersubstantivalists should allow for the possibility of regions ofspace that are not also material objects. So which premises do they reject? Not thepremises about essential properties: presumably supersubstantivalists agree that itis essential to regions of space (and so to material objects) that they be spatiallyextended. They should deny the (suppressed) premise that material objects are a‘kind of thing,’ things that can be characterized in terms of their essential proper-ties. Instead, say that material objects differ from other regions of spacetime in theiraccidental properties—like, for example, mass and charge.

Of course this reply is not open to dualists; dualists do not think materialobjects differ from regions of space only accidentally. So (turning now to my firstquestion), how do dualists resist this argument?

Dualists should reject (1). Descartes’s argument for (1), as it occurs in hisdiscussion of the stone in Principles 2:11, is to cast about for a counterexample andfail to find one.9 He thinks that for any other property some material object has,

8One quick objection to this argument is to say that material objects can bepointsized, and so have no spatial extent at all. But in that case the same is pre-sumably true of regions of space—if there can be pointsized material objects therecan be pointsized regions—and the argument could be modified to take this intoaccount. Descartes would not accept this new argument, though, since he thoughtit impossible for a material object to lack all spatial extent. And this plays a crucialrole in his argument against atoms in 2:20.

9In 2:4 Descartes argues that no perceptible properties, like hardness and color,are essential to material objects. Something similar goes on in Descartes’s discus-

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there can be a material object that lacks it; and so no other property can be essentialto material objects.10

A dualist might say that mass is an essential property of material objects.(Descartes, of course, lived before any such property was recognized.) But this can’tbe right, because there are massless particles. Bennett (2001, pp.30-32), followingLocke and others, objects that Descartes fails to rule out solidity, or impenetrability,as an essential property of bodies. But I’m not sure that impenetrability is essentialto material objects. Neutrinos may have some power to exclude other particlesfrom occupying just the region they do, but it is very weak (that is why they are sohard to detect); do we really think it is impossible for there to be a kind of particlethat is completely uncoupled from other kinds, and does not interact with them atall? In any case, dualists have a better reason to reject (1). They should say thatit is essential to material objects that they occupy regions of space.11 But it is notessential to regions of space that they occupy regions of space. So these two kindsof spatially extended thing do not share all their essential properties.

Descartes’s argument for supersubstantivalism is not very good, but I thinkthere is a better argument for supersubstantivalism that is Cartesian in spirit.

5 Spatiotemporality

Descartes thought there couldn’t be two kinds of spatial thing. I think that reflectionon what it is to be a spatial (or, better, spatiotemporal) thing can still give us somereason to be supersubstantivalists, even if not the reason Descartes thought.

A virtue of supersubstantivalism, to the eyes of a metaphysician, is that itprovides the materials for a better account of spatiotemporality than does dualism.This gives us some reason to be supersubstantivalists.

sion of the wax in the second meditation; but there his primary aim is to argue thatthe nature of the wax ‘is in no way revealed by my imagination, but is perceived bythe mind alone’ (page 85).

10For most of the passage he argues that for any property a stone has, that verystone could lack it. But when he argues that heaviness is not essential to the stone,he cites the fact that fire (which he regards as a material body) is not heavy.

11Markosian (2000) defends something like this view.

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What is it to be a spatiotemporal thing? What is it to be extended in spaceand time? Supersubstantivalists have a simple answer: to be spatiotemporal is to bea region of spacetime, to be part of spacetime.

What do dualists say? They will say that to be spatiotemporal is either to be aregion of spacetime, or to bear the occupation relation to some region of spacetime.But this view has several disadvantages.

First, there is a problem about essences. It is essential to material objectsthat they be spatiotemporal. For dualists this means: for anything at all, if it occu-pies some region of spacetime, then it is necessary that it occupies some region ofspacetime. But this is to accept a necessary connection between distinct existences.Supersubstantivalists do not need to accept this necessary connection.

Second, there is a problem of inclusion. Which things are spatiotemporal?Material objects certainly are. But some material objects are parts of other materialobjects; and composite material objects are located where their parts are. That is,it is necessary that if x is part of y and x occupies region L then y also occupiesL.12 Why is this necessary? Dualists can give no answer. They cannot explainthe necessary connection between parthood and occupation. Supersubstantivalistsdo not believe in an occupation relation, and so have no necessary connection toexplain. (Or, better, supersubstantivalists do believe in an occupation relation, butthey deny that it is a fundamental relation, and instead analyze it in terms parthood:a region of spacetime occupies just those regions that are parts of it. On this account,there is no mystery in the necessity that material objects occupy those regions thattheir parts occupy.13)

12L need not be the largest region y occupies. Some might accept a weakercondition than the one in the text. They might say that it is necessarily only that if xis part of y and x occupies L then y occupies a region that contains L. The differencebetween these two does not affect my argument.

13I’m oversimplifying the way we use ‘occupies’ and our talk of locations andplaces generally. Supersubstantivalists will say that sometimes we use ‘occupies’ toexpress the converse of the parthood relation; sometimes we use it the way relation-alists think we always use it: to express some complicated relation to other materialobjects, as when I stand on the subway platform and say, ‘every morning I standin the same place and wait for the train’ to mean something like ‘every morning Istand next to this wall and wait for the train.’ This will come up again below.

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Third, there is a problem of exclusion. Some philosophers want to expandthe realm of the spatiotemporal. Penelope Maddy, for example, claims that setsare spatiotemporal. She uses this claim to defend the existence of sets by arguingthat we can see sets, and so that we have empirical evidence for their existence.14

It is easy for supersubstantivalists to reject Maddy’s claim. They say that onlyregions of spacetime are spatiotemporal, and no set is a region of spacetime. Itis not so easy for dualists. They must argue that sets cannot bear the occupationrelation to regions of spacetime. But it is open to someone like Maddy to say:dualists admit a necessary connection between occupation and parthood. I admit anadditional necessary connection between occupation and membership: sets occupyjust those regions their members occupy. I cannot explain why there should bethis connection; but neither can dualists explain why there should be a connectionbetween occupation and parthood.

6 Newtonian Supersubstantivalism

Descartes hoped to derive supersubstantivalism from premises we should all ac-cept. That’s an ambitious project. There is a less ambitious way to argue forsupersubstantivalism. Instead of arguing that material objects must be regions ofspacetime, we argue that they can be. Regions of spacetime can do all the workwe need material objects to do. For the sake of simplicity, then, we should identifymaterial objects with regions of spacetime.15

Newton took this approach. His manuscript ‘De Gravitatione’ contains an at-tempt to describe the nature of material objects.16 He ends up describing a possibleworld in which, he claims, it would be appropriate to say that certain regions ofspace are (also) material objects. Newton proceeds in stages.

First he asks us to imagine the following scenario. Suppose the dualism istrue, that material objects really are distinct from spacetime. God chooses someregion of empty space—call it ‘Joe.’ Whenever a material object is about to occupy

14(Maddy 1990), chapter 2, especially pp.58-59.15(Sider 2001) and (Bennett 1984) also give this motivation for super-

substantivalism.16(Newton 1962). Howard Stein discusses this manuscript in his (1970).

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Joe, God exerts a force on that object in the direction away from Joe sufficient toprevent it from occupying Joe. So when they get close enough to Joe, superballsbounce right back; photons are reflected; and beer bottles shatter. Over time Godmay change which region he watches over in this way; we may assume that hechanges it continuously and that all the regions have the same shape. (We maybe tempted to say, as God varies which region of space enjoys his favor, that Joeis moving. But this is just a misleading appearance; Joe is a region of space, andNewton denies that regions of space can move.)

Of course even in Newton’s physics material objects act on each other in waysother than by contact; but we can easily imagine God also causing each materialobject to be accelerated towards Joe, that acceleration being indirectly proportionalto the square of the distance between that object and Joe (or the distance betweenthat object’s center of mass and some point that falls inside Joe). Material objects inthis world would move, then, just as if there were some material object occupyingJoe and attracting them gravitationally.

Of this scenario Newton asserts, ‘it seems impossible that we should not con-sider this space [that is, Joe] to be truly body from the evidence of our senses....forit will be tangible on account of its impenetrability, and visible, opaque and coloredon account of the reflection of light, and it will resonate when struck because theadjacent air will be moved by the blow’ (139).

Now it is doubtless true that were we in such a world, confined to exploringit with our senses, we would conclude that Joe is ‘truly body’; but since we arenot so confined, but know more directly the metaphysical truth of the situation, itis clear that Joe is no material object. A world in which God behaves in the waydescribed is a world in which God deceives some of his creatures. (Descartes wouldbe appalled.)

The problem is that while some regions of space—Joe, in particular—aremasquerading as material objects, there are also, in the world, real material objects—that is, material-objects-according-to-dualists. Our use of ‘material object’ is not sopermissive that it could apply to both regions of space and things distinct from re-gions of space in the same world (considered as actual). There’s an obvious way toavoid this problem: Newton next asks us to imagine a world that contains none of

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what dualists call ‘material objects,’ but instead contains only space, certain regionsof which God has chosen to watch over as he watched over Joe in the previous sce-nario. (Of course, now watching over a given region involves ensuring that no otherchosen region overlaps it, rather than ensuring that no dualist material object occu-pies it.) In such a world, Newton (perhaps) asserts—he declines to wholeheartedlyendorse the thesis—supersubstantivalism is true.17

Newton’s assertion is implausible, even to someone sympathetic to super-substantivalism. In the first scenario, the one in which there were also dualist ma-terial objects, there was something there for God to repel away from his chosenempty region of space. Without those things, though, in what way do the chosenregions of space differ from the unchosen regions? They do not differ in their in-trinsic properties, since it is no part of God’s activity (as Newton has described it)to endow the chosen regions with special intrinsic properties. Nor does God endowthese regions of space with the disposition to repel each other. For I take it thattwo things that differ in their dispositional properties must also differ intrinsically.Intuitively, a bar of lead does not become as fragile as glass just because a magicianresolves to cast a spell causing it to shatter whenever anyone strikes it with a lightblow; God, though more powerful, no more alters the dispositions of the regions ofspace than the magician alters the dispositions of the bar.

So the special regions differ from the ordinary ones only in their relationalproperties: some regions have, and others lack, the property of being chosen. Surelythere must be more of a difference than this between regions which are also materialobjects and regions which are not. It can seem that all we have here is a worldcontaining no material objects, only empty space, in which God has chosen to playa game with himself whereby he ‘outlines’ certain regions of space and changeswhich regions are outlined according to certain definite rules.18

I am not sure why Newton wanted God to do so little for the regions of spacehe has chosen to be material objects. Why have God merely watch over them, whenGod could give them the power to watch over themselves, by endowing them withcertain intrinsic properties? Perhaps the following idea motivated Newton: all that

17That is, supersubstantivalism is true and there are material objects.18(Bennett 1984) makes a similar complaint.

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really matters, when we describe a supersubstantivalist world, is how the regionsinteract with other things; as long as they interact in the right sort of way, it doesn’tmatter what they are like ‘in themselves.’19

7 Moderate Supersubstantivalism

Newton’s strategy, again, is to argue that regions of space can do all the work weneed material objects to do. I object to the way he implemented this strategy: ma-terial objects do more than just prevent other material objects from occupying theregions they occupy. Let’s keep the strategy and change the implementation.

The physical theories that have been considered fundamental since Newtonpostulate two kinds of material objects: particles and fields. What work do particlesand fields do in these theories? I will start with the particles. Particles are theprimary bearers of intrinsic properties like mass and charge; and they move around.

A moderate version of supersubstantivalism can allow regions of spacetime toinstantiate mass and charge. So that bit of work can be done by regions of spacetimeas easily as by material objects. But what about motion? Can a region of spacetimemove?

Versions of supersubstantivalism that treat space and time as separate things—like Newton’s—have trouble here. According to these versions, material particlesare regions of space. Normally substantivalists say that a particle moves if it occu-pies different regions of space at different times. But this won’t work if particlesjust are regions of space; whatever sense we give ‘occupies’ in this context, a regionof space cannot occupy different regions of space at different times.

This kind of supersubstantivalist might do better by adopting a relationalistaccount of motion: a region moves just in case its distance from some other regionchanges in time. But it seems impossible for regions of space to move in this sense,either: it seems necessary that the distance between any two points of space bethe same at all times. There is some conceptual room to maneuver around this

19Something like this line of thought occurs on page 140. Others haveshared Newton’s motivating idea that dispositions don’t require categorical bases—Faraday (1965) inclined toward supersubstantivalism for similar reasons, andBlackburn (1993) gives the idea a more general defense.

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impossibility, but the moves aren’t attractive. I’ll mention them, just to set themaside.

We might give up mereological essentialism for regions of space, and saythat one and the same region can include different points at different times. Theneven if the distance between any two points of space is the same at all times, thedistances between regions can change. Or we might accept mereological essential-ism, deny that particles really move, and try to explain away appearances to thecontrary. Different regions are particle-like (by instantiating mass properties and soon) at different times; since the pattern of mass-instantiation changes continuously,it looks like one persisting thing is moving around, but this is an illusion.20

Luckily, we can avoid having to say either of these things by denying thatspace and time are separately exiting things. Believe instead in a four-dimensionalspacetime that unifies them. In this context dualists explain what it is for a particleto move by citing features of its worldline. So, for example, a particle acceleratesjust in case its wordline is not straight. Supersubstantivalists who identify a particlewith its worldline can say the same thing.

So much for particles. What about fields? The case for identifying the partsof physical fields (like the electromagnetic field) with regions of spacetime is evenbetter than the case for identifying particles with regions of spacetime. In this casewe don’t have to worry about motion: whether and how the electromagnetic fieldmoves is of no importance to the theory of electromagnetism.

Of course this was not always so. According to classical electromagnetism,light is a wave in the electromagnetic field. And many in the 19th century assumedthat if there is a wave then something must be waving: something must be changingits shape, as the water in the pond changes shape when a rock falls in it. So therehad to be some physical stuff that carried electromagnetic waves: the ether.

I think that even this theory can be given a supersubstantivalist interpretation:identify each part of the ether with its worldline in four-dimensional spacetime, and

20This is analogous to what mereological essentialists who aren’t supersubstan-tivalists say: strictly speaking, there are no persisting tables, but it looks like thereare because different things are momentarily table-like in the same place at differenttimes.

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explain its motion in the same way the motions of particles are explained. But theether interpretation of electromagnetism has been discredited. We no longer thinkthat there can be a wave only if something physical is changing shape, and so weno longer need a moving ether.

8 The Modal Argument Against Supersubstantivalism

As I said in the last section, supersubstantivalism is best combined with a belief infour-dimensional spacetime, rather than in space and time separately. This versionof supersubstantivalism entails the doctrine of temporal parts: every persisting thingis composed of instantaneous temporal parts. Arguments against the doctrine oftemporal parts, then, are also arguments against supersubstantivalism. But I havenothing to add here to the debates over the doctrine of temporal parts.

The most serious remaining argument against supersubstantivalism is themodal argument:

(1) I could have been three feet to the left of where I actually am right now.

(2) No region of spacetime could have been three feet to the left of where itactually is right now.

(3) So I am not a region of spacetime.

This argument should look familiar: it is structurally similar to the problem of mo-tion for supersubstantivalists that I discussed above. There the problem is account-ing for the apparent fact that one and the same material object can be in differentplaces at different times; here the problem is accounting for the apparent fact thatone and the same material object can be in different places in different possibleworlds. For this reason, parts of my reply to the modal argument will resembleparts of my discussion of the problem of motion.

Supersubstantivalists could deny the first premise and try to explain away ourtendency to accept it. They could say: a region three feet to the left of me could havebeen intrinsically just like I am right now. Maybe we are confusing this possibilitywith the possibility according to which I am three feet to the left.

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This move doesn’t look very plausible. What, then, about denying the secondpremise? Before looking at how to deny it, we need to clarify its meaning. Whatdoes talk of where a region of spacetime actually is mean? Supersubstantivalistsshould insist it means something relational: for it to be possible that a region ofspacetime be three feet from where it actually is at some time, there must be apossible world in which its distances from other regions at that time are differentfrom what they actually are. So if region A is actually one foot to the left of regionB at that time, then in the other possible world A must be four feet to the left of B,and so on.

Is this really possible? There are two ways in which this possibility mightbe realized. First, the points in A might be different distances from the points inB in the two possible worlds. Second, while all points of spacetime stand in thesame geometrical relations to each other in both possible worlds, the points thatcompose A and B differ in the two worlds. So to deny (2) supersubstantivalistsmust give up one of two doctrines. They must either reject (some version of) geo-metrical essentialism, the doctrine that points of spacetime have their geometricalproperties essentially; or reject compositional essentialism for spacetime regions,the doctrine that regions of spacetime cannot contain different points in differentpossible worlds.21

I am happy to give up geometrical essentialism, for reasons I discussed in 2: itdoes not sit well with general relativity. To supersubstantivalists who do accept bothof the above doctrines I offer the following reply to the argument. They should saythat our modal talk is context dependent. In some context the essentialist doctrinesare true; in some context they are false. In the first kind of context, then, (1) is falseand (2) is true. In the second kind, (1) is true and (2) is false. In no context are bothpremises true: when we are led to think both are true, it is because the context isshifting back and forth.

One way to explain this context dependence is to give a counterpart-theoreticanalysis of these de re modal claims. In some contexts (contexts in which we are

21This is the doctrine that deserves the name ‘mereological essentialism forspacetime regions;’ but this name is already widely used for the claim that regionscannot contain different points at different times.

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thinking about regions of spacetime as regions of spacetime) we use a counter-part relation that values geometrical similarity and makes these essentialist claimstrue. In other contexts (contexts in which we are thinking about regions of space-time as material objects) we use a counterpart relation that values other kinds ofsimilarity—like similarity with regard to mass and charge distribution—over geo-metrical similarity, and makes the essentialist claims false.22

22By ‘values geometrical similarity’ I mean more than that counterparts are geo-metrical duplicates; I mean also that counterparts of two regions stand in the samegeometrical relations as the regions do.

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Chapter 4

What Makes Time Different FromSpace?

1 Introduction

No one denies that time and space are different; and it is easy to catalog differencesbetween them. I can point my finger toward the west, but I can’t point my fingertoward the future. If I choose, I can now move to the left, but I cannot now chooseto move toward the past. And (as D. C. Williams points out) for many of us, ourattitudes toward time differ from our attitudes toward space. We want to maximizeour temporal extent and minimize our spatial extent: we want to live as long aspossible but we want to be thin.1 But these differences are not very deep, and don’tget at the essence of the difference between time and space. That’s what I want tounderstand: I want to know what makes time different from space. I want to knowwhich difference is the fundamental difference between them.

I will argue for the claim that (roughly) time is that dimension that plays acertain role in the geometry of spacetime and the laws of nature. (In this paper,then, I focus on what is distinctive about time, and say little about what is distinc-tive about space.) But before giving the argument I want to put my question inslightly different terms. Instead of asking, ‘what makes time different from space?,’

1(Williams 1951, page 468). Williams actually says that we care how long welive but do not care how fat we are.

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I want to ask, ‘what makes temporal directions in spacetime temporal, rather thanspatial?’2 After rejecting some bad answers to this question I’ll present my view.

2 Spacetime Diagrams and Directions in Spacetime

It is often helpful, when approaching problems in physics and in metaphysics, todraw a spacetime diagram. Spacetime diagrams represent the careers in space andtime of some material objects. Traditionally in a two-dimensional spacetime dia-gram (the easiest kind to draw on paper) the horizontal axis represents space and thevertical axis represents time. So suppose I’m confined to one dimension in space:I can only move to the left or to the right. Then the diagram in figure 2 might rep-resent part of my career. The zig-zag line represents me; it’s my worldline. (I’mincorrectly represented as point-sized, but that’s not important.) According to thediagram I stand still for a while; then I walk to the left, stop, stand still for a littlewhile longer, and then walk back to the right.

I said I wanted to ask what makes temporal directions in spacetime temporal,rather than spatial. So what is a direction in spacetime? We can use spacetimediagrams to get a sense for what directions in spacetime are. To represent a directionin spacetime (at some spacetime point) on the diagram we can draw an arrow, orvector, on the diagram at the point that represents that point of spacetime. So infigure 2 the arrow labeled ‘A’ points in the leftward direction in space and the arrowlabeled ‘B’ points in the future direction in time. There are in this diagram, then, atleast two temporal directions: toward the future and toward the past; and two spatialdirections: toward the left and toward the right.

Two arrows may point in the same direction while being of different lengths.A direction then is an equivalence class of vectors—the set of all vectors that point

2It is important to distinguish this question from another commonly discussedquestion. Many philosophers want to know what makes the future different fromthe past. But that is not what I am asking. Toward the future and toward the past areboth temporal directions, and I am not asking what makes one temporal directionthe direction toward the future and the other, the direction toward the past. InsteadI’m asking, what makes either of them a temporal, rather than spatial, direction inthe first place?

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Figure 4.1:

Time

Space

in the same direction and differ only in their length. Following standard usage, Iwill sometimes call a vector that points in a temporal direction a ‘timelike vector,’and a vector that points in a spatial direction a ‘spacelike vector.’

What about the arrows labeled ‘C’ and ‘D’? They don’t seem to point in eithera temporal or a spatial direction. What to say about arrows like C and D reallydepends on what geometrical structure the spacetime represented by the diagramhas. In (two-dimensional) neo-Newtonian spacetime every arrow that does not pointeither to the left or the right points in a temporal direction, while in Minkowskispacetime (the spacetime of the special theory of relativity) arrows that are less

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Figure 4.2:

Time

Space

A

B C

D

than 45◦ from the vertical (like C) point in a temporal direction, while arrows thatare more than 45◦ (like D) from the vertical point in a spacelike direction.

Why frame the discussion in terms of spatial and temporal directions, ratherthan space and time? Modern physical theories are formulated in terms of a four-dimensional spacetime, instead of in terms of three-dimensional space and one-dimensional time separately. In some older theories (Newtonian mechanics, in par-ticular) there is a way to identify certain regions of spacetime as points of spaceand other regions as instants of time. But in more recently theories, especially the

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Figure 4.3: Newtonian Spacetime

instant of timepoint of space

general theory of relativity, this cannot always be done.We can identify points of space and instants of time with certain regions of

Newtonian spacetime because Newtonian spacetime has certain special geometri-cal features (see figure 4.3). There is a unique and geometrically preferred wayto divide up this four-dimensional spacetime manifold into a sequence of three-dimensional Euclidean submanifolds. Each of these Euclidean submanifolds iswell-suited to play the role we want instants of time to play: events that occuron the same submanifold occur simultaneously. So these submanifolds are instantsof time. A point of space, then, is a line in spacetime perpendicular to each time.(Events located on the same line, whether simultaneous or not, occur in the same

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place.) Facts about which regions are points of space, and which are instants oftime, are absolute: not relative to any observer or frame of reference.

In Newtonian spacetime the distinction between time and space and betweentemporal and spatial directions coincide. A vector that points in a spatial directionis one that points along (is tangent to) a time, and so points toward other points ofspace. A vector that points in a temporal direction is one that points at an angle toa time, and so points in the direction of future or past times.

But the special geometrical features that allow us to identify points of spaceand instants of time with certain regions of Newtonian spacetime are missing inother spacetimes. In neo-Newtonian spacetime, though there is a non-relative wayto identify regions of spacetime with times, there is not a non-relative way to iden-tify regions of spacetime with points of space. Different inertial observers willregard different events as occurring in the same place, and so will regard differentlines in spacetime as points of space, without the geometry of spacetime privilegingone of them over the others.3 This happens to time as well in Minkowski spacetime,the spacetime of special relativity.

In general relativity it gets worse. According to this theory the geometry ofspacetime varies from world to world, depending on the distribution of matter ineach world. In some of those worlds spacetime can be divided up into instantsof time and points of space. (As before the times are three-dimensional submani-folds having certain geometrical properties, but in general relativity they need nothave a Euclidean geometry. Points of spaces are curves in spacetime (they neednot be straight lines) that run oblique to each time and meet some further geomet-rical conditions. Typically there are many ways to divide spacetime up, just as inMinkowski spacetime, none more preferred by the geometry than the others.) Butin other worlds spacetime cannot be divided in this way at all. (Gödel’s solutionis an example.) In these worlds no regions of spacetime count as points of spaceor instants of time, not even relative to some inertial observer. But even in theseworlds there is a distinction between the temporal and the spatial aspects of space-

3I said above that in Newtonian spacetime a point of space is a line in spacetimethat intersects each time at a right angle. In neo-Newtonian spacetime we can nolonger say which lines intersect each space at a right angle and which do not.

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time, because we can still distinguish spacelike from timelike directions. (Given apoint on the worldline of a conscious observer in one of these spacetimes, for exam-ple, we can still tell in which directions from that point his future mental episodeslie.) Since the distinction between directions is more general than the distinctionbetween space and time, it is the distinction on which I want to focus.

3 Does Geometry Distinguish Temporal from Spatial Directions?

The view that the distinction between temporal and spatial directions is a geometri-cal one is a natural one to take when one studies spacetime theories. Those theoriesattribute one or another geometrical structure to spacetime, and when a given the-ory is explained, the distinction between temporal and spatial directions is usuallyexplained in geometrical terms.

For example, according to the special theory of relativity, the geometry ofspacetime satisfies the axioms of Minkowski geometry. That geometry allows usto assign lengths to vectors in spacetime. And the lengths of vectors that point intemporal directions have a different sign than the lengths of vectors that point inspatial directions. One kind has negative lengths; the other, positive.

We might hope to explain what makes temporal directions temporal by ap-pealing to the signs of the lengths of vectors that point in those directions; but Idon’t see how this approach could work, for three reasons.

First, while looking at the signs of the lengths of the vectors may allow us topick out two disjoint classes of vectors (namely, the class of vectors with negativelengths, and the class of vectors with positive lengths), it doesn’t allow us to figureout which of these the class of vectors that point in temporal directions. For it is amatter of convention whether vectors that point in temporal directions get assignedpositive or negative lengths; different textbooks adopt different conventions. Andthe distinction between temporal and spatial directions is not a conventional one.

Second, a difference in the signs of their lengths is too formal and abstract tobe the fundamental difference between vectors that point in temporal and those thatpoint in spatial directions, and so between time and space.

And third, distinguishing between the two kinds of directions in terms of the

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signs of vectors’ lengths only works in spacetimes with Minkowski geometry. Novectors have negative lengths in neo-Newtonian spacetime, but we do not want tosay that no directions in that spacetime are temporal. Of course in that spacetimethe geometrical distinction between spatial and temporal directions is explaineddifferently. Roughly speaking, in neo-Newtonian spacetime we have two ways toassign (positive) lengths to vectors. We are told that vectors that point in timelikedirections are those that have non-zero length according to a particular one of thoseways. (The same is true in Newtonian spacetime.) But it won’t do to say that whatmakes a timelike vector timelike is that is satisfies the following condition: eitherit is a vector in Minkowski spacetime and it has (say) negative length, or it is avector in neo-Newtonian spacetime and it has positive length according to one par-ticular metric (and so on with clauses for each different spacetime). For temporaldirections in a world with one spacetime geometry have something in common withtemporal directions in a world with some other spacetime geometry. And an answerto the question, ‘in virtue of what are temporal directions temporal?’, must tell ussomething about what they have in common. Even if the current proposal correctlydistinguishes temporal from spatial directions, it doesn’t say what temporal direc-tions have in common. So it is not the answer we’re looking for.

While we have not yet found a geometrical way to distinguish timelike fromspacelike directions, we do have a geometrical way to distinguish directions thatare either spacelike or timelike from those that are neither. It is a matter of conven-tion whether we say that timelike vectors in Minkowski spacetime have positive ornegative lengths. But it is not a matter of convention that they (along with spacelikevectors) have non-zero length. And we can establish that vectors with zero lengthare neither spacelike or timelike on geometrical grounds alone. The zero vector,which points in no direction at all, has zero length, but in some spacetimes othervectors do as well. In particular, ‘lightlike’ vectors in Minkowski spacetime havezero length. (These vectors point along possible paths of light rays.) Whicheverspacetime geometry we look at, that geometry assigns lengths to vectors as a wayto assign distances, either spatial or temporal, between spacetime points. (Thislength is determined by adding up (really, integrating) the lengths of vectors tan-gent to a certain path between the two points.) But adding up a bunch of zeros just

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gives zero, so adding up the lengths of vectors with zero length couldn’t be a wayto determine the temporal or spatial distance between two points.

We have, then, a geometrical way to divide the vectors in any spacetime intothe class of vectors that are either spacelike or timelike, and the class of vectors thatare neither. And we also have a geometrical way to divide the class of vectors thatare either spacelike or timelike into two subclasses. (In Minkowski spacetime (andin the spacetimes of general relativity as well) we divide them into the subclasswith negative lengths and the subclass with positive lengths. In Newtonian andneo-Newtonian spacetime we divide them into the subclass with positive lengthaccording to one way of assigning lengths, and the subclass with positive lengthsaccording to the other way of assigning lengths. (Let’s say that two vectors thatbelong to the same subclass are ‘of the same kind.’ Talk of vectors that are of thesame kind, then, is reserved for vectors that are either spatial or temporal.)) But wedon’t yet have a way to designate one of those subclasses as the class of vectors thatpoint in temporal directions.

4 Dimensionality

If we’re looking for a geometrical way to distinguish temporal from spatial direc-tions, dimensionality considerations are probably our best bet. In four-dimensionalMinkowski geometry, whichever convention about the signs of lengths one uses,time ends up being one-dimensional and space ends up being three-dimensional.Perhaps it is because it is one-dimensional that time is time.

Before examining this thesis I’ll say something about what it means to saythat time is one-dimensional.

Intuitively speaking, to say that time is one-dimensional is to say that we canrepresent time as a line, and that all events that occur in time can be assigned aposition on that line.4

As I said above in section 2, in Newtonian and neo-Newtonian spacetimethere is a unique geometrically preferred way to slice up the four-dimensional

4Circles are one-dimensional too, so strictly speaking time could be one-dimensional even if we had to represent time as a circle.

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spacetime into a one-dimensional sequence of three-dimensional Euclidean sub-manifolds. Each three-dimensional submanifold is a time, and the sequence givestheir temporal ordering. Since every event occurs somewhere in spacetime, ev-ery event occurs somewhere in this one-dimensional sequence. So time is one-dimensional in the intuitive sense in these spacetimes because it divides up in thisway.

In Minkowski spacetime there are many geometrically preferred ways to sliceup the four-dimensional spacetime into a sequence of three-dimensional Euclideansubmanifolds. We can still, then, temporally order all events on a line, even thoughthere is no unique way to do so. (There are pairs of events x and y such that x occursbefore y according to one slicing but y occurs before x according to another slicing.)

But we do not want to say in general that time is one-dimensional just in casespacetime divides up naturally into a one-dimensional sequence of three-dimensionalsubmanifolds. For, as I mentioned, some general relativistic spacetimes cannot bedivided up into a sequence of three-dimensional submanifolds. And there is still a(somewhat technical) sense in which time is one-dimensional in worlds containingthose spacetimes.

The vectors in spacetime at any given spacetime point form a four-dimensionalvector space; the maximum dimension of a subspace containing only timelike vec-tors (and the zero vector) is one, while the maximum dimension of a subspacecontaining only spacelike vectors (and the zero vector—I’ll leave this implicit fromnow on) is three.5 Time is one-dimensional in this more technical sense not just ingeneral relativistic spacetimes, but also in the other spacetimes I’ve mentioned; sothis more technical sense is a generalization of the one I gave earlier.

To see why this is so in Minkowski spacetime, consider (for ease of visualiza-tion) three-dimensional Minkowski spacetime, depicted in figure 4. From a givenpoint in that spacetime the set of points that can be reached by light rays forms a

5Here’s a brief, intuitive explanation of vector spaces. Think of a (three-dimensional) vector space as the set of all arrows that can be drawn (in ordinaryspace) from a point. The arrow of zero length counts: it’s the zero vector. A sub-space of that vector space is a subset of the arrows such that either all of the arrowslie in the same plane (that’s a two-dimensional subspace) or all of the arrows lie onthe same line (that’s a one-dimensional subspace).

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Figure 4.4: Minkowski Spacetime

future-directed light cone timelike

vector

double cone: the future and past light cones at that point. The set of vectors that lieinside either light cone are the timelike vectors. If you pick any line through thatcone and consider the set of vectors that point along that line, then (as I explainedin footnote 5) those timelike vectors form a one-dimensional subspace of the spaceof all vectors at that point. But there couldn’t be a two-dimensional subspace oftimelike vectors: if there were, then there would have to be some plane through thatpoint in spacetime such that all the vectors at that point that lie in that plane alsofall inside either the past or future light cone. But no plane lies entirely inside thetwo light cones.

It’s not controversial that time is one-dimensional in the familiar spacetimetheories. The controversial claim is that it is dimensionality that makes timelikevectors timelike. To be explicit, the controversial claim is:

Dimension: A timelike vector ~v is one that satisfies the following: the maximum

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dimension of a subspace containing only vectors of the same kind as~v is one.6

I objected above to the first way of using geometry to distinguish spatial fromtemporal directions that it didn’t tell us what temporal directions in different space-times had in common. Dimension does better, because it does say something aboutwhat they have in common.

There are, however, several problems for Dimension. First of all, it entails thatit is necessary that time is one-dimensional. Now, I do not want to deny that timeis necessarily one-dimensional. (Some philosophers have tried to describe worldsin which time has two dimensions, but I don’t find those descriptions convincing.7)But I am not sure I want to affirm it, either. It is better to have a view that does notrule it out.

Second, the dimensionality difference between time and space doesn’t seemdeep enough to be the fundamental difference between the two. I take it that thefundamental difference between space and time will illuminate the other, less fun-

6In Minkowski spacetime the maximum dimension of a subspace containingonly null vectors (lightlike vectors and the zero vector) is one. But Dimension doesnot entail that these vectors are timelike. In section 3 I argued that null vectors arenot timelike on geometrical grounds alone; Dimension, like the other principles Iwill later advance, is only meant to determine which of the remaining vectors aretimelike. (Recall I said in section 3 that talk of two vectors being of the same kindis reserved for vectors that (unlike null vectors) are either spatial or temporal.)

7Judith Thompson tries to describe such a world in her (1965). (This exampleis discussed in more detail in (MacBeath 1993).) It’s a world in which two peopledisagree about the temporal order of certain pairs of events. Each person thinks heoutlives the other. The argument that time is two-dimensional in this world goes,in outline, like this. (1) There is no way to temporally order events on a line sothat both people are right. (2) But both are equally well-placed to determine thetemporal order of events, so we do not want to say that one is right and the otherwrong about the temporal order. So (3) the events cannot be temporally ordered ona line; time has more than one dimension.

But this conclusion does not follow. Time can be one-dimensional even if thereis no uniquely correct way to temporally order events on a line. In Minkowskispacetime, for example, two observers can disagree about the temporal order ofevents. But time is still one-dimensional in that spacetime.

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damental, differences between them, and help us explain those other differences.But the difference in dimensionality doesn’t do this.

Third, and most important, it seems possible that both time and space be one-dimensional. But in a possible world with two-dimensional Minkowski spacetime,all the (non-null) vectors meet the condition in Dimension. So Dimension entailsthat all (non-null) directions in that world are temporal, and so (since no directionis both spatial and temporal) that that world contains no spatial directions at all. Butthat can’t be right: surely it’s possible that special relativity be true and that timeand space both be one-dimensional.

The problem is that while vectors of one kind satisfy the condition in Dimen-sion, vectors of the other kind do as well; while we have already established thatall timelike vectors are of the same kind. Someone who maintains that dimension-ality is the only factor that does any work to distinguish timelike from spacelikedirections, then, must admit that the condition in Dimension is necessary but notsufficient for a direction to be timelike; and that no condition is sufficient in everyworld. He might then revise his view as follows:

Dimension∗: If a vector is timelike then the maximum dimension of a subspacecontaining only vectors of the same kind as it is one; all timelikevectors are of the same kind; and to the extent that these conditionsfail to determine which vectors are timelike, it is to that extent inde-terminate which vectors are timelike.

In worlds where just one kind of vector satisfies the condition in Dimension, then,Dimension and Dimension∗ agree that vectors of that kind are timelike. But in worldswith two-dimensional spacetimes Dimension∗ entails that it is indeterminate whichkind of vector is timelike.

In the end, though, this move to indeterminacy fails. It fails not becauseI insist that it must be perfectly determinate in every world which directions aretimelike. But surely in some two-dimensional worlds, complicated worlds in whichplenty is happening, there is a fact of the matter about which directions are temporal.So we should reject Dimension∗ as well as Dimension.

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I have looked at two ways to distinguish temporal from spatial directions ingeometrical terms, and found reasons for rejecting both. Might some other attemptto distinguish them in geometrical terms succeed where these have failed? Theanswer is ‘no.’ The two-dimensional spacetimes that make trouble for Dimensionalso make trouble for any attempt to use geometry alone to distinguish temporalfrom spatial directions. For the roles that timelike and spacelike directions playin the geometry of two-dimensional Minkowski spacetime (and two-dimensionalNewtonian spacetime) are symmetric. Since timelike and spacelike directions playsymmetric roles in the two-dimensional spacetime geometries, any attempt to dis-tinguish temporal from spatial directions by isolating a geometrical role that onebut not the other plays is bound to fail.

So what else other than or in addition to the geometry of spacetime makes thedifference between spacelike and timelike directions?

5 Laws of Nature

Timelike and spacelike directions play different roles in the laws of physics that wehave taken seriously as the fundamental laws governing our world. Those laws gov-ern the evolution of the world in timelike directions, but not in spacelike directions.

This claim might look analytic (‘of course evolution happens in time’), butI’m using ‘govern the evolution of the world’ in a stipulated sense that does notbuild time into its meaning. The laws govern the evolution of the world in somedirection just in case the laws, together with information about what is going on insome region of spacetime, yield information about what is going on in regions ofspacetime that lie in that direction from the initial region.

My meaning can be made more precise using an example. Earlier I said thatthe spacetimes of Newtonian mechanics and special relativity, as well as some ofthe spacetimes of general relativity, can be partitioned into a sequence of times—a sequence of three-dimensional submanifolds. Now in Newtonian gravitationaltheory, given information about what is going on on one time, the laws determinewhat is going on on the rest of the times.8 These laws govern the evolution of the

8I am pretending here (for purposes of illustration) that Newtonian gravitational

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world from one time to the others. And a similar claim is true for other laws we’veconsidered fundamental.

Now, timelike vectors are not tangent to any time, on any way of partitioningany given spacetime into times. Rather, no matter which partitioning of spacetimeinto times you use, timelike vectors point from one time toward others. So timelikevectors point in the directions in which the laws govern the evolution of the world.9

The same is not true of points of space. If I know what is going on right here(at this location in space) for all time, the laws tell me nothing about what is goingon anywhere else at any time. The laws do not govern the evolution of the world inspacelike directions.

I used the laws of Newtonian gravitational theory as an example, and theselaws are deterministic. If some laws are deterministic, then given information aboutthe state of the world on some time, those laws yield complete information aboutthe state of the world on other times. But not all possible laws are deterministic;on some interpretations the laws of quantum mechanics are an example of indeter-ministic laws. But even here there is a difference between the roles timelike andspacelike directions play in the laws: given information about the state of the worldat a time, these laws assign probabilities to possible states at other times; but theydo not do so for points of space.

(There is another role that timelike directions play in some familiar laws thatspacelike directions do not: quantities like mass, charge, and energy are conservedin timelike directions, but not in spacelike directions. But when there are conser-vation laws like this, they are usually derived from the dynamical laws (as, forexample, the law of conservation of charge follows from Maxwell’s equations—thedynamical laws for electromagnetism). So I need not explicitly mention this as a

theory is deterministic, even though there are good arguments that it is not. Earmandiscusses these arguments in his (1986).

9As I’ve said, not every general relativistic spacetime can be partitioned intotimes. But the laws of general relativity still govern the evolution of worlds withthose spacetimes in timelike directions at a local level: some four-dimensional re-gions of those spacetimes can be partitioned into times, and (if the region is theright shape) the laws determine what is happening at all times given informationabout what is happening at one time.

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second role in the laws that distinguishes timelike from spacelike directions.)It is not controversial that timelike directions play these roles in the laws with

which we are most familiar. I propose that we take these roles as constitutive: what

it is to be a timelike direction is to play these roles in the laws. To be explicit, theproposal is this:

Laws: Any direction (that is either spatial or temporal according to the geometry)in which the laws govern the evolution of the world is a timelike direction.

Let me make two remarks about this proposal.First, I do not claim that it is necessary that the laws of nature govern the

evolution of the world in some direction(s) in spacetime.10 I do not claim, that is,that it is necessary that there be some timelike role in the laws to be filled. Perhapsthere are possible worlds with strange laws of nature in which there is no such role.But I do claim that in such worlds no direction is a timelike direction.

Second, my proposal presupposes that the laws of nature are more fundamen-tal than the distinction between timelike and spacelike directions. It presupposes,roughly, that it is possible to state the laws of nature without using the words ‘time’and ‘space.’ For if in order to state the laws we had to presuppose that time andspace had already been distinguished, then it would be going in a circle to then ap-peal to the laws to distinguish time from space. This is not a problem, though. Westandardly state laws without appealing to the distinction between time and space.In formal presentations of, say, Newtonian gravitational theory, the distinction be-tween timelike and spacelike directions is always made in some informal remarksafter the author has described the geometrical structure of spacetime and beforehe writes down the equations of the theory. But the equations can be understoodperfectly well with the informal remarks removed.

Or, to take a more concrete case, consider Newton’s first law: a body notacted on by any forces moves with a constant velocity. ‘Velocity’ means the sameas ‘change in spatial location with respect to change in temporal location.’ I claimthat we can re-write this law to remove the reference to space and time. We can dothis by using quantifiers: first, replace ‘velocity’ with ‘change in location along the

10I’m using ‘the laws of nature’ non-rigidly.

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Figure 4.5:

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x direction with respect to change in location along the y direction’; then, prefacethe laws with the quantificational phrase, ‘there are two (distinct) directions, x andy, such that x and y play such-and-such geometrical roles, and....’

6 Testing Our Intuitions

Let’s take a look at a particular world with a two-dimensional spacetime and seeif we agree that the laws distinguish space from time even though the geometry ofspacetime does not. Consider the spacetime diagram in figure 4.5.

This diagram depicts the distribution of matter in spacetime in some possibleworld. (Suppose that spacetime has the familiar Newtonian structure, so that talkof space and time makes sense.) Normally we read the vertical axis of spacetimediagrams as the time axis. I ask you to forget about that convention for now and

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suppose you do not know which axis represents time. I contend that if you do notknow what the laws are, you are unable to tell which axis is time. And that isevidence that it is the laws that are doing the work to make one axis time.

It certainly seems that there are possible worlds correctly represented by thisdiagram in which the vertical axis represents time; and also possible worlds cor-rectly represented by this diagram in which the horizontal axis represents time. Tomake this plausible, let me describe for you one world of each kind. Call the worldin which the vertical axis represents time ‘the vertical world,’ and the other, ‘thehorizontal world.’ First, a description of the vertical world:

Vertical: there are two particles that accelerate toward each other, untilthey meet in an elastic collision and rebound back the way they came;then they slow down, turn around, and accelerate back toward eachother, repeating this cycle for all time.

And here’s a description of the horizontal world:

Horizontal: for a long time nothing happens. Then an infinite num-ber of pairs of particles come into existence; one member of each pairmoves off to the right, the other to the left. Each particle bounces off aparticle coming from the other direction, then is annihilated in a colli-sion with the particle with which it was created. Then nothing happensfor the rest of time.

By itself the diagram doesn’t favor one of these descriptions over the other.Things are different, I think, when I tell you what laws of nature govern the

world the diagram depicts. Let the laws be Newton’s three laws of mechanics anda slightly amended version of Newton’s law of universal gravitation.11 Of courseNewton’s laws contain terms like ‘velocity’ and ‘acceleration’ which are defined in

11According to Newton’s law the force between two bodies is inversely propor-tional to the square of the distance between them. As the distance between twobodies goes to zero, the force gets infinitely large. A natural way to extend New-ton’s laws to deal with this case is to have the particles bounce off each other in aperfectly elastic collision.

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terms of space and time; suppose them to be redefined in the way I mentioned aboveon page 87, so that velocity ends up being change in position along the horizontalaxis with respect to change in position along the vertical axis.12 So in these lawsthe vertical axis plays the time role in the definition of ‘velocity,’ and also plays therole that I described above: the laws govern the evolution of the world along thevertical axis. It seems to me that when we add these laws to the description of theworld depicted by the diagram, we know enough to know that the vertical axis isthe time axis, and we rule out Horizontal as a correct description of that world.

Let me address two worries one might have about this example. First, onemight worry that my strategy here—maintaining that knowing only how matter isdistributed in spacetime, without knowing what the laws are, leaves us unable to tellwhich axis is time—is incoherent if a broadly empiricist theory of laws of natureis true. According to such theories, the laws of nature supervene on the occurrent(categorical, non-dispositional) facts. Since the occurrent facts surely include factsabout the structure of spacetime and the distribution of matter in it, an empiricistwill say that by fixing the facts about matter and spacetime I’ve already fixed whatthe laws are, and so already fixed which directions are timelike.

This may be so; I won’t take a stand here on whether some empiricist theoryof laws is true. But even if an empiricist theory is true, it is coherent to ask youto remain ignorant about the laws even though you know about the distribution ofmatter in spacetime, for it is not always obvious what the laws are in a world with

12So where the original laws contained ‘...velocity...’ the new laws will say ‘thereare two (distinct) directions, x and y, such that x and y play such-and-such geomet-rical roles, and....change in position in the x direction with respect to change inposition along the y direction...’

Given that the horizontal and vertical axes play symmetric geometrical roles,you might wonder why it is the direction along the horizontal axis, rather than thedirection along the vertical axis, that gets to play the time-role in the definitionof velocity. But there is no answer to this question. The way the laws are statedguarantees that (in two-dimensional worlds with Newtonian spacetime) one kindof direction will play the time role in the definition of velocity and the other willnot; but it is undetermined which—there are other worlds with the same spacetimegeometry and the same laws in which the other plays that role.

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a given set of occurrent facts. And the fact that we cannot tell which axis is timewhen ignorant of the laws but can when we know the laws is still evidence that it isthe role they play in the laws that makes timelike directions timelike.13

Second, one might worry about the descriptions in Vertical and Horizontal.Those descriptions seem to contain more information about the distribution of mat-ter in spacetime than the spacetime diagram alone does. And the points at whichthe descriptions go beyond the information contained in the diagram are points atwhich the descriptions disagree. The spacetime diagram tells us only which space-time points are occupied by material objects. The descriptions contain further in-formation about how many material objects there are, and which points each objectoccupies. In Vertical I said that there were two particles, while in Horizontal Isaid there were infinitely many. Now, I claimed that knowing only how matter wasdistributed in spacetime isn’t enough to allow us to figure out which axis is time.But one might complain that by presenting the spacetime diagram I hadn’t givencomplete information about how matter is distributed in spacetime. Complete in-formation requires the kind of information contained in Vertical and Horizontal.If I had said that there are only two particles, and each one occupies the points onjust one of the curvy lines, then perhaps it would have seemed obvious that thevertical axis is time.

To allay this worry let me make some further stipulations about the world(s)represented by the diagram. I’ll add information so Vertical and Horizontal won’tcontain additional information.

Suppose that in worlds represented by the diagram there are uncountablymany material objects, that some of them are mereologically simple (without properparts), and that each simple object occupies exactly one of the occupied spacetimepoints. Suppose also that for any collection of the simple objects there is a mere-ologically complex object that they compose. (In slogans, then, I’m saying that

13An empiricist theory of laws entails that at most one of Vertical and Horizon-tal describes a possible world. For the descriptions disagree about which axis istime, and so (on my view) disagree about the laws, but agree on the occurrent facts.Empiricists should read the possibilities at work in my argument as epistemic, ratherthan metaphysical, possibilities.

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four-dimensionalism and the doctrine of unrestricted mereological composition aretrue in these worlds.) Vertical and Horizontal seem to disagree about how manythings there are and which regions those things occupy, but we can suppose that thisis mere appearance. These descriptions are not complete; they don’t tell the storiesof the spatiotemporal careers of every material object. Instead they only tell the sto-ries of a few salient ones. When we move from Vertical to Horizontal we switchwhich axis we regard as the time axis; doing this brings about a shift in which ofmaterial objects are salient.

So far I’ve presented my proposal and given it some intuitive support. I willnow discuss an objection to it.

7 Symmetric Laws and Indeterminacy

The claim that geometry distinguishes timelike from spacelike directions ran intoproblems with spacetimes in which timelike and spacelike directions play sym-metric roles. I said that there is still a difference between timelike and spacelikedirections in some of those worlds, and that it is their different roles in the lawswhich distinguishes them. But what about worlds in which timelike and spacelikedirections play symmetric roles in both the geometry and the laws? There isn’t anyreason to deny that such laws are possible; there are even examples of such laws.The wave equation for a wave in one dimension, for example, is

1v2

∂2φ

∂t2 =∂2φ

∂x2

(φ is a function on spacetime; it tells you, intuitively speaking, ‘how high’ the waveis at each point of spacetime.) Now v here is the speed of the wave, and we’re freeto choose units in which it’s 1. In that case, the equation becomes

∂2φ

∂t2 =∂2φ

∂x2

It is clear that in this law time and space play symmetric roles: switching t and x

leaves the equation the same. Moreover, the roles time and space play in these laws

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both fit the description I gave above: these laws govern the evolution of the waveboth along the time axis and along the space axis.

Laws entails that all directions in this world are timelike, and that is not right.There are three ways to respond to this problem. First, we could conclude thatwe do not yet have a complete account of what makes time different from space,and we could search for some other feature of the world that is doing work todistinguish spacelike from timelike directions. Second, we could conclude thatit is just a brute fact that the timelike directions in this world are timelike, thatthere is nothing informative to be said about what makes them timelike. Or third,(paralleling the move from Dimension to Dimension∗) we could conclude that if thegeometry and laws fail to single out one kind of vector as the timelike vectors, thenit is indeterminate which directions are timelike.

We should choose the third alternative. It does not seem that it could be justa brute fact which directions are timelike; and in these highly symmetric worlds itis hard to see what else, other than the geometry and the laws, could distinguishtimelike from spacelike directions. Worlds governed by the wave equation look thesame no matter which axis we regard as the time axis.

To deal with these symmetric worlds, then, amend Laws as follows:

Laws∗: If a direction is timelike then the laws govern the evolution of the worldalong it; all vectors that point in timelike directions are of the same kind;and to the extent that these conditions fail to determine which vectors aretimelike, it is to that extent indeterminate which vectors are timelike.

Laws and Laws∗ agree on all worlds except worlds like the one governed by the one-dimensional wave equation. In such worlds Laws∗ entails that it is indeterminatewhich kind of vector is timelike.

How bad is it to admit that it is sometimes indeterminate which directionsare timelike? Certainly it’s perfectly determinate in our world which directions aretimelike. Earlier I claimed that there are some two-dimensional worlds—complicatedones in which there is a lot going on—in which it is perfectly determinate whichdirections are timelike. But I don’t think that this must be perfectly determinate inevery two-dimensional world, so I see no problem accepting that it is indeterminate

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which directions are timelike in worlds in which time and space play symmetricroles in both the laws and the geometry of spacetime.

8 Laws That Govern in a Spacelike Direction

I turn now to a second objection. My view entails that it is not possible that therebe laws that govern the evolution of the world is a spacelike direction. But (so theobjection goes), surely this is possible. Surely we can produce examples of possibleworlds with this feature.

I have yet to hear a convincing example. Let me go through some of theexamples I have heard.14

Example 1: In this world, it is a law that to the left of every apple isan orange, and to the left of every orange is an apple. (Suppose we’vefixed, once and for all, which direction in space is to the left.) This lawgoverns in a spacelike direction: if I know that there is an apple here, Iget information about what is going on to the left.

Example 2: In this world, there is a special plane dividing space inhalf; and it is a law that the contents of space on one side of the planeare perfectly mirrored on the other side. This law governs in a spacelikedirection: if I know that there is a red sphere in a certain place on oneside of the plane, I get information about what is going on at the verysame time in the corresponding place on the other side of the plane.

I am enough of a metaphysician to take examples like these somewhat seriously.I have a three-part response. First, I don’t think the laws in these examples aredoing enough to govern the evolution of the world in a spacelike direction. Giveninformation about what is going on at one place for all time, they do not give infor-mation about what is going on everywhere else at all times. These laws only give

14I heard many of these examples at the APA Eastern Division meeting, 2004.Among those who suggested examples are Eric Lormand, James Van Cleve, PhilipBricker, and Jonathan Schaffer. There were others, as well, and I can no longerremember who suggested which examples.

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information about what is going on in some other places. (The law in example 1gives information about places to the right of, and some of the places to the left of,the initial place. The law in example 2 gives information about just one other placefor all time: the place that is the mirror-reflection of the initial place.) So they’re notdoing in a spacelike direction what laws like Newtonian mechanics do in a timelikedirection.

Second, even if there is a world in which it is true that to the left of everyapple is an orange (and so on), and a world with mirror symmetry, I’m not surewhy I should believe that it would be a law that to the left of every apple is anorange, or that the world exhibits mirror symmetry. (Certainly the law about applescouldn’t be a fundamental law, but maybe it could be derivative.) For I find it hardto have intuitions about what the laws of some world are, given descriptions of thegoings-on in those worlds.

Of course, some philosophers can argue that if there is a world in which tothe left of every apple is an orange (and so on), and the world is simple enoughin other ways, then it is a law that to the left of every apple is an orange. Theseare philosophers who accept the best system theory of laws. According to the bestsystem theory of laws, those true statements are laws that are theorems of the de-ductive system that best balances simplicity and strength.15 If the apple world issimple enough in other ways, then the statement that to the left of every apple is anorange might make it in to the best system, and so might be a law.

Earlier I tried to remain neutral between primitivist theories of laws and em-piricist theories of laws (like the best system theory). Now it appears that I musttake sides. I reject the argument in the previous paragraph because I reject the bestsystem theory of laws.

This is not just an ad hoc move, though. I do not just reject the theory becauseif conflicts with my theory of the difference between space and time. My primaryreason for rejecting it is that it fails to respect our modal intuitions about lawhood.Briefly, empiricism about laws (and so the best system theory in particular) entailsthat there cannot be worlds that differ merely in what laws govern them. But I accept

15(Lewis 1986b).

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the counterexamples opponents of the best systems theory offer to this claim.16 Forexample, it seems that there is a possible world in which an empty spacetime hasa Minkowski geometry, and in which special relativity is true; and another worldin which an empty spacetime has a Minkowski geometry, and in which generalrelativity is true. (In the latter world, had there been any matter, spacetime wouldhave been curved; this is not true in the former world.) Empiricists about laws mustsay that at least one of these worlds is not possible.

There is a third part of my response to examples like 1 and 2. It is part ofthe description of the possible worlds in these examples that certain directions arespacelike and certain others, timelike. But doesn’t this beg the question? Whyshould we accept that the directions called ‘spacelike’ in the description really arespacelike?

One might refuse to answer this question. But then I am not sure why Ishould take seriously the claim that there are possible worlds like those describedin the examples. Or, one might answer this question by proposing an alternativetheory of what makes time different from space. I’ll criticize a few such theoriesin the next section. Or, one might say: we can imagine the worlds described, andit is part of the content of our imagining that the directions called ‘spacelike’ arespacelike. And since imagination is a guide to possibility, that gives us reason tobelieve that these worlds are possible.

How does this act of imagination work? Maybe like this: we imagine watch-ing the history of the world in question unfolding, as if we were watching a movie.We can tell which directions are timelike and which spacelike when watching amovie (without needing any theory of the difference to help us); in the same way,we can tell which directions are timelike and which spacelike in these worlds.

But imagining watching the history of the world unfold as if watching a movieis only a legitimate way to learn which directions in that world are timelike if thoseare the experiences an observer who existed in the world would have. But if theworlds described in these examples contain conscious observers, then there mustbe many more laws than the ones mentioned. Once all these laws are also takeninto account, I doubt it will be so clear that the laws govern the evolution of the

16For example, in (Carroll 1994).

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world in a spacelike, not a timelike, direction.

Example 3: In this world, the laws are those of Newtonian mechanicsand Newton’s law of universal gravitation. This law governs in a space-like direction: if I know that there is a particle here of a certain massnow, I know something about the value the gravitational field now hasat every other point.

This example is not convincing. First, it is not clear that on a correct interpreta-tion, Newtonian gravitational theory say that there really is any such thing as thegravitational field. And even if it does, knowing that there is now a particle hereof a certain mass doesn’t tell me the value of the field at any other point; to knowthat I’d have to know how many particles there were in total, and what their massesand positions are right now. If I know anything about what is going on elsewhere,it is only how the particle here contributes to the value of the gravitational fieldelsewhere; but since this is consistent with the field’s actual value being anything atall, and since we get no information of the kind we’re really looking for (namely,information about which other regions of space are occupied by material objects),these laws are not really governing the world in a spacelike direction.

Example 4: In this world, the laws of quantum mechanics govern theworld. In an EPR-type experiment, there are two particles some dis-tance apart, and if we measure the spin on one of them in some direc-tion, we know with certainty the outcome of a measurement of the spinof the other particle in that same direction, even if the measurementevents are spacelike-separated. So these laws govern the evolution ofthe world in a spacelike direction.

This example is not convincing, for two reasons. First, I want to complain that, asin examples 1 and 2, the determination by laws in a spacelike direction here is notthe robust kind needed to refute my view. But there is a second, more important,problem. For laws to govern the evolution of the world in a spacelike direction,it must at least be the case that given information about what is going on in oneplace (say, right here), the laws give information about what is going on at other

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places that are spacelike separated from it. But if all we know is that (after beingmeasured) some particle right here has spin-up in some direction, the laws don’ttell us anything about what is going on elsewhere. They only give us informationif we also know that there is another particle somewhere else, and that the ‘system’comprising the two particles is in an entangled state. But this information is not justinformation about what is going on right here.

9 Alternative Views

Finally, I will review some alternative views about what makes time different fromspace, and say why I reject them. My goal is not to give these theories detailedformulations and refutations; I aim only to suggest why I think they go in the wrongdirection from the start.

9.1 The Causal Theory of Time

According to this view, a direction is timelike just in case it is a possible direc-tion of causation. This theory was inspired, I think, by a certain way of thinkingabout Minkowski spacetime. There is a synthetic axiomatization of this space-time’s geometry using just one two-place predicate that can be taken to mean ‘x

and y are causally connectible.’ Perhaps this axiomatization is getting the meta-physics right: the spatiotemporal relations in Minkowski spacetime, and so factsabout which events occur before which other events, are derived from a more fun-damental relation of causal connectibility.17

I reject this theory for two reasons. The first, and less important reason, isthat it precludes the possibility of instantaneous causation. It looks like Newtonianmechanics involves instantaneous causation—according to that theory the sun’s be-ing a certain distance from the earth instantaneously causes the earth to experiencea certain force—and I accept that Newtonian worlds are metaphysically possible.But I place more importance on a second reason. I just don’t think that facts aboutcausation are more fundamental than the difference between space and time. Butthey have to be, for this theory to be right.

17See (Sklar 1974).

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(My theory, of course, requires that the laws be more fundamental than thedifference between space and time. One might wonder why I am comfortable withthis and uncomfortable with the causal theory of time. I won’t give a detailedanswer to this question; I will just point out that many contemporary philosophersshare the feeling that causation is less fundamental than lawhood.18)

9.2 Three-Dimensionalism

Three-Dimensionalism is the view that material objects persist through time withouthaving temporal parts. Since it is commonly admitted that material objects areextended in space by having spatial parts, there is an asymmetry here between spaceand time. One might try to distinguish space from time using this asymmetry: timeis that dimension in which material objects are extended without being made up ofparts.

Three-Dimensionalism is controversial, so insofar as I am not a three-dimensionalistI am not tempted by this proposal. But I do not think three-dimensionalists shouldbe either. One standard argument that material objects that persist through timemust have temporal parts is the argument from temporary intrinsics: if somethingis spherical at a time, then it must have a temporal part that is spherical simpliciter,on pain of making sphericality a relation to times, and not an intrinsic property atall.19 Three-dimensionalists think they can resist this argument. But if they can,then they can also resist the parallel argument that material objects that are spatiallyextended must have spatial parts, the argument from local intrinsics: if somethingis red in one place and green in another, then it must have a spatial part that is redsimpliciter, on pain of making redness a relation to places. (Maybe redness andgreenness are not the best examples, but there are others.) So three-dimensionalistsought to admit the possibility of material objects that are spatially extended without

18David Lewis (1986a) is one example: he analyzes causation in terms of coun-terfactuals, and his truth-conditions for counterfactuals appeal to laws. But evenphilosophers who reject counterfactual analyses of causation, like Maudlin (2004),believe that laws are more fundamental than causation.

19This argument is much discussed; it is presented, among other places, in (Lewis1986b) and (Sider 2001).

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having any spatial parts. But once that possibility is granted, there is no longer anasymmetry between the way material objects are (or can be) extended in space andthe way they are (or can be) extended in time.

9.3 The Brute Fact View

According to the brute fact view, there is no need to appeal to geometry or the lawsto distinguish spacelike from timelike directions. Instead, there is no way to dis-tinguish spacetime from timelike directions in other terms. There is no informativeanswer to the question, ‘what makes timelike directions timelike?’

One way to put this view is to say that there is a fundamental property, theproperty of being a timelike direction. But it might seem odd to believe that thingslike directions could have fundamental properties. If we confine our attention toNewtonian spacetime we can avoid this oddness by avoiding talk of directions.Some regions of this spacetime are times, and others are not. The ones that aretimes have some fundamental intrinsic property that the others do not: the propertyof being a time. End of theory.

This theory is not plausible. Certainly the regions that have this special prop-erty must also play the appropriate role in the geometry. (Supposing we characterizeNewtonian spacetime using two distance functions, the role is as follows: each re-gion contains all and only the points that are zero distance from any point in thatregion, according to one of those distance functions.) But why is there this neces-sary connection between this special property and a certain geometrical role? Thebrute fact view gives no answer. My view does better: since it does not postulatethe special property, it has no necessary connection to explain.

But maybe that is just a caricature of the brute fact view. Here is a closelyrelated view one might have. Do not postulate a special non-geometrical funda-mental property of being a time. Instead, pick out one of the geometrical relationsthat gives spacetime its structure, and make it special. Sticking to my focus onNewtonian spacetime, one way to put the view is as follows: this spacetime get itsstructure, let us suppose, from two distance functions. One of these is the temporal

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distance function, and the other is the spatial distance function.20 Times are regionscontaining all and only the points that are zero distance from any point in them,according to the temporal distance function. What makes one function the temporaldistance function, rather than the spatial distance function? There is no answer. Itis just a brute fact.

This view I take more seriously as a competitor to my own than the othersI have considered. But I do think it is wrong. The reason appeals, again, to two-dimensional Newtonian spacetime.

Take a world w with two-dimensional Newtonian spacetime, and ‘rotate’ allthe matter in that world ‘90 degrees’21 to produce a new world w∗. I can describethis world in a bit more detail, but I don’t want to beg any questions by callingcertain regions of spacetime in w∗ ‘times’ or ‘points of space.’ So I will have topick out regions of spacetime in w∗ using features those regions have in w. Thedescription ‘regions of spacetime that are points of space in w’ picks out a definiteset of regions of spacetime in w∗, while leaving it open whether those regions arealso points of space in w∗. In more detail, then, w∗ looks like this: events that aresimultaneous in w occur (in w∗) in regions that (in w) lie on different times butare in the same place. The laws of w∗ are also obtained from the laws of w by‘rotation’: the w∗-laws treat regions that are points of space in w as the w-laws treatthe regions that are instants of time in w. I think w∗ is qualitatively indiscerniblefrom w. But the brute fact view cannot say this. According to the brute fact view,two-dimensional Newtonian spacetime is not symmetric in this way. So the brutefact view entails that either the ‘rotation’ operations cannot be performed (that is,there is no such ‘rotated’ world), or, if they can, they result in a world that is veryvery different, qualitatively, from w. So I reject this view.

Both versions of the brute fact view are similar to the view that what makesthe future different from the past is that the future direction in time has some special

20There are analogous ways to formulate the view on other accounts of the fun-damental spatiotemporal relations that characterize Newtonian spacetime.

21Of course, this doesn’t really make sense in the geometry of Newtonian space-time. What I really mean is: consider this world represented on a two-dimensionalEuclidean plane, like a piece of paper; then rotate everything 90 degrees on thisEuclidean plane; now consider the world this new diagram represents.

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intrinsic property that the past direction in time lacks.22 I reject the brute fact viewfor the same reason many reject this view about the difference between the pastand the future.23 In the later case, it seems that a world in which the distributionof matter were ‘mirror reversed’ around a given time (the laws, being time-reverseinvariant (we may suppose), would be the same) would not be a world in whicheverything were ‘going backwards,’ but a world in which the direction that is actu-ally the future direction is the past direction. Many who hold this view identify thefuture direction with the direction in which entropy increases;24 so it is not intrinsicto the future direction that it be the future direction. I accept this view about the dif-ference between the future and the past. And my view about the difference betweenspace and time is analogous.

22(Maudlin 2002) seems to defend this view, as does (Earman 1974).23(Price 1996) is an example.24(Reichenbach 1956) is one example.

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Bibliography

Adams, Robert (1979). ‘Primitive Thisness and Primitive identity.’ Journal of

Philosophy 76(1): 5–26.

Barbour, Julian (1999). The End of Time. New York: Oxford University Press.

Belot, Gordon (1995). ‘New Work for Counterpart Theorists: Determinism.’ British

Journal for the Philosophy of Science 46: 185–195.

— (2000). ‘Geometry and Motion.’ British Journal for the Philosophy of Science

51: 561–595.

Bennett, Jonathan (1984). A Study of Spinoza’s Ethics. Hackett.

— (2001). Learning from Six Philosophers, volume 1. New York: Oxford Univer-sity Press.

Blackburn, Simon (1993). ‘Filling In Space.’ In Essays in Quasi-Realism, chap-ter 14. New York: Oxford University Press.

Brighouse, Carolyn (1994). ‘Spacetime and Holes.’ PSA: Proceedings of the Bien-

nial Meeting of the Philosophy of Science Association 1: 117–125.

— (1997). ‘Determinism and Modality.’ British Journal for the Philosophy of

Science 48: 465–481.

Butterfield, Jeremy (1989). ‘The Hole Truth.’ The British Journal for the Philoso-

phy of Science 40: 1–28.

103

— (2002). ‘Critical Notice: Julian Barbour, The End of Time.’ British Journal for

the Philosophy of Science 53: 289–330.

Carroll, John W. (1994). Laws of Nature. New York: Cambridge University Press.

Clifford, William Kingdon (1882). ‘On the Space-Theory of matter.’ In RobertTucker (ed.), Mathematical Papers, 21–22. London: Macmillan.

Descartes, Rene (1988). Selected Philosophical Writings. New York: CambridgeUniversity Press.

Earman, John (1974). ‘An Attempt to Add a Little Direction to ‘The Problem of theDirection of Time’.’ Philosophy of Science 41(1): 15–47.

— (1986). A Primer on Determinism. Boston: D. Reidel Publishing Company.

— (1989). World Enough and Space-Time. Cambridge, MA: MIT Press.

— (1995). Bangs, Crunches, Whimpers, and Shrieks. New York: Oxford UniversityPress.

Earman, John and John Norton (1987). ‘What Price Substantivalism? The HoleStory.’ The British Journal for the Philosophy of Science 38: 515–525.

Faraday, Michael (1965). ‘A speculation touching Electric Conduction and theNature of Matter.’ In Experimental Researches in Electricity, volume 2, 284–293. New York: Dover.

Field, Hartry (1980). Science Without Numbers. Princeton: Princeton UniversityPress.

— (1989). ‘Can We Dispense With Space-time?’ In Realism, Mathematics and

Modality, chapter 6. New York: Basil Blackwell.

Forbes, Graeme (1987). ‘Places as Possibilities of Location.’ Nous 21: 295–318.

Friedman, Michael (1983). Foundations of Space-Time Theories. Princeton: Prince-ton University Press.

104

Harman, Gilbert (1986). Change in View. Cambridge, MA: M.I.T. Press.

Hoefer, Carl (1996). ‘The Metaphysics of Space-Time Substantivalism.’ Journal of

Philosophy 93(1): 5–27.

Kripke, Saul (1980). Naming Necessity. Cambridge, MA: Harvard University Press.

Leibniz, G.W. and Samuel Clarke (2000). Correspondence. Ed. Roger Ariew.Indianapolis: Hackett.

Lewis, David (1968). ‘Counterpart Theory and Quantified Modal Logic.’ Journal

of Philosophy 65: 113–126.

— (1983). ‘New Work for a Theory of Universals.’ Australasian Journal of Phi-

losophy 61: 343–377.

— (1986a). ‘Causation.’ In Philosophical Papers, volume 2, 159–171. New York:Oxford University Press.

— (1986b). On The Plurality of Worlds. New York: Blackwell.

MacBeath, Murray (1993). ‘Time’s Square.’ In Robin Le Poidevin and MurrayMacBeath (eds.), The Philosophy of Time, 182–202. New York: Oxford Univer-sity Press.

Maddy, Penelope (1990). Realism in Mathematics. New York: Oxford UniversityPress.

Markosian, Ned (2000). ‘What are Physical Objects?’ Philosophy and Phenomeno-

logical Research 61: 375–395.

Maudlin, Tim (1990). ‘Substances and Space-Time: What Aristotle Would HaveSaid to Einstein.’ Studies in History and Philosophy of Science 21: 531–561.

— (1993). ‘Buckets of Water and Waves of Space: Why Spacetime Probably is aSubstance.’ Philosophy of Science 60: 183–203.

— (2002). ‘Remarks on the Passing of Time.’ Proceedings of the Aristotelian

Society 102: 259–274.

105

— (2004). ‘Causation, Counterfactuals, and the Third Factor.’ In John Collins, NedHall and L. A. Paul (eds.), Causation and Counterfactuals, chapter 18, 419–444.Cambridge, MA: MIT Press.

McCormmach, Russell (1970). ‘H. A. Lorentz and the Electromagnetic View ofNature.’ Isis 61(4): 459–497.

Melia, Joseph (1998). ‘Field’s Programme: Some Interference.’ Analysis 58(2):63–71.

— (1999). ‘Holes, Haecceitism and Two Conceptions of Determinism.’ British

Journal for the Philosophy of Science 50: 639–664.

Newton, Isaac (1962). ‘De Gravitatione.’ In Marie Hall and Alfred Hall (eds.), Un-

published Scientific Papers of Isaac Newton, 121–156. Cambridge: CambridgeUniversity Press.

O’Leary-Hawthorne, John and J. A. Cover (1996). ‘Haecceitism and Anti-Haecceitism in Leibniz’s Philosophy.’ Nous 30(1): 1–30.

Pooley, Oliver and Harvey R. Brown (2002). ‘Relationalism Rehabilitated? I. Clas-sical Mechanics.’ British Journal for the Philosophy of Science 53: 183–204.

Price, Huw (1996). Time’s Arrow and Archimedes’ Point. New York: OxfordUniversity Press.

Quine, W. V. (1981). ‘Things and Their Place in Theories.’ In Theories and Things,chapter 1. Cambrige, MA: Harvard University Press.

Reichenbach, Hans (1956). The Direction of Time. Los Angeles: University ofCalifornia Press.

Royden, H. L. (1959). ‘Remarks on Primitive Notions for Elementary Euclideanand non-Euclidean Plane Geometry.’ In Leon Henkin (ed.), The Axiomatic

Method with Special Reference to Geometry and Physics, 86–96. Amsterdam:North-Holland.

106

Sider, Theodore (2001). Four-Dimensionalism. New York: Oxford UniversityPress.

Sklar, Lawrence (1974). Space, Time, and Spacetime. Berkeley: University ofCalifornia Press.

Stein, Howard (1970). ‘On The Notion of Field in Newton, Maxwell, and Beyond.’In Roger H. Stuewer (ed.), Minnesota Studies in the Philosophy of Science, vol-ume 5, 264–287. Minneapolis: University of Minnesota Press.

— (1977). ‘Some Philosophical Pre-History of General Relativity.’ In J. Earman,C. Glymour and J. Stachel (eds.), Minnesota Studies in the Philosophy of Science,volume 8. Minneapolis: University of Minnesota Preess.

Tarski, Alfred (1959). ‘What is Elementary Geometry?’ In Leon Henkin (ed.),The Axiomatic Method with Special Reference to Geometry and Physics, 16–29.Amsterdam: North-Holland.

Thomson, Judith Jarvis (1965). ‘Time, Space, and Objects.’ Mind 74(293): 1–27.

Wheeler, John A. and Edwin F. Taylor (1963). Spacetime Physics. 1st edition. SanFrancisco: W. H. Freeman and Company.

Williams, Donald C. (1951). ‘The Myth of Passage.’ The Journal of Philosophy

48(15): 457–472.

Wilson, Mark (1993). ‘There’s a Hole in the Bucket, Dear Leibniz.’ Midwest

Studies in Philosophy 18: 202–241.

107